-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon.{v₁, u₁} C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
have h : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X) P.X) Q.X ⟶
    coequalizer (M.actRight ▷ (N.tensorBimod (P.tensorBimod Q)).X)
      (Iso.hom (α_ M.X W.X (N.tensorBimod (P.tensorBimod Q)).X) ≫ M.X ◁ (N.tensorBimod (P.tensorBimod Q)).actLeft))
  (Iso.hom (α_ M.X N.X P.X) ▷ Q.X ≫
      Iso.hom (α_ M.X (tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X) Q.X) ≫
        CategoryStruct.comp (Y :=
          tensorObj.{v₁, u₁} (𝒞 := inst) M.X
            (tensorObj.{v₁, u₁} (𝒞 := inst)
              (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)) Q.X))
          (M.X ◁ coequalizer.π (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft) ▷ Q.X :
            tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X) Q.X) ⟶
              tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                (tensorObj.{v₁, u₁} (𝒞 := inst)
                  (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)) Q.X))
          (CategoryStruct.comp (Y :=
              tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                (coequalizer (TensorBimod.actRight N P ▷ Q.X)
                  (Iso.hom (α_ (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)) Y.X Q.X) ≫
                    coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft) ◁ Q.actLeft)))
              (M.X ◁
                  coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
                    (Iso.hom
                        (α_ (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)) Y.X Q.X) ≫
                      coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft) ◁ Q.actLeft) :
                tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                    (tensorObj.{v₁, u₁} (𝒞 := inst)
                      (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)) Q.X) ⟶
                  tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                    (coequalizer (TensorBimod.actRight N P ▷ Q.X)
                      (Iso.hom
                          (α_ (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)) Y.X Q.X) ≫
                        coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft) ◁ Q.actLeft)))
              (CategoryStruct.comp (Y :=
                  (parallelPair (X :=
                        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                          (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                            (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                        (Y :=
                        tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                          (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                            (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                        (M.actRight ▷
                            coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                              (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft) :
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                              (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ⟶
                            tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                              (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                        (Iso.hom
                              (α_ M.X W.X
                                (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                  (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft))) ≫
                            M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q) :
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                              (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ⟶
                            tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                              (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))).obj
                    WalkingParallelPair.one)
                  (M.X ◁
                      coequalizer.desc
                        (CategoryStruct.comp (Y :=
                          coequalizer (X :=
                            tensorObj.{v₁, u₁} (𝒞 := inst)
                              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X) P.X) Q.X)
                            (N.actRight ▷ P.X ▷ Q.X) ((Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft) ▷ Q.X))
                          (Iso.inv
                            (PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
                              (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)))
                          (coequalizer.desc
                            (Iso.hom (α_ N.X P.X Q.X) ≫
                              CategoryStruct.comp (Y :=
                                tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                                (N.X ◁ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft) :
                                  tensorObj.{v₁, u₁} (𝒞 := inst) N.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) ⟶
                                    tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                                      (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                                (coequalizer.π
                                    (N.actRight ▷
                                      coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))
                                    (Iso.hom
                                        (α_ N.X X.X
                                          (coequalizer (P.actRight ▷ Q.X)
                                            (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))) ≫
                                      N.X ◁ TensorBimod.actLeft P Q) :
                                  tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                                      (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
                                    coequalizer (X :=
                                      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                                        (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                                      (Y :=
                                      tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                                        (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                                      (N.actRight ▷
                                          coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft) :
                                        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                                            (coequalizer (P.actRight ▷ Q.X)
                                              (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
                                          tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                                            (coequalizer (P.actRight ▷ Q.X)
                                              (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                                      (Iso.hom
                                            (α_ N.X X.X
                                              (coequalizer (P.actRight ▷ Q.X)
                                                (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))) ≫
                                          N.X ◁ TensorBimod.actLeft P Q :
                                        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                                            (coequalizer (P.actRight ▷ Q.X)
                                              (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
                                          tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                                            (coequalizer (P.actRight ▷ Q.X)
                                              (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))))
                            sorry))
                        sorry :
                    tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                        (coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
                          (Iso.hom (α_ (N.tensorBimod P).X Y.X Q.X) ≫ (N.tensorBimod P).X ◁ Q.actLeft)) ⟶
                      tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                        (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                          (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                  (colimit.ι
                      (parallelPair (X :=
                        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                          (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                            (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                        (Y :=
                        tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                          (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                            (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                        (M.actRight ▷
                            coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                              (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft) :
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                              (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ⟶
                            tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                              (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                        (Iso.hom
                              (α_ M.X W.X
                                (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                  (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft))) ≫
                            M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q) :
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                              (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ⟶
                            tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                              (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft))))
                      WalkingParallelPair.one :
                    (parallelPair (X :=
                            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                              (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                            (Y :=
                            tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                              (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                            (M.actRight ▷
                                coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                  (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft) :
                              tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                                  (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                    (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ⟶
                                tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                                  (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                    (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                            (Iso.hom
                                  (α_ M.X W.X
                                    (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                      (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft))) ≫
                                M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q) :
                              tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                                  (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                    (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ⟶
                                tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                                  (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                    (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))).obj
                        WalkingParallelPair.one ⟶
                      colimit
                        (parallelPair (X :=
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                            (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                              (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                          (Y :=
                          tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                            (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                              (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                          (M.actRight ▷
                              coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft) :
                            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                                (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                  (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ⟶
                              tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                                (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                  (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                          (Iso.hom
                                (α_ M.X W.X
                                  (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                    (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft))) ≫
                              M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q) :
                            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                                (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                  (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ⟶
                              tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                                (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                  (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft))))) :
                (parallelPair (X :=
                        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                          (coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
                            (Iso.hom (α_ (N.tensorBimod P).X Y.X Q.X) ≫ (N.tensorBimod P).X ◁ Q.actLeft)))
                        (Y :=
                        tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                          (coequalizer (TensorBimod.actRight N P ▷ Q.X)
                            (Iso.hom
                                (α_ (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)) Y.X
                                  Q.X) ≫
                              coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft) ◁ Q.actLeft)))
                        (M.actRight ▷
                            coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
                              (Iso.hom (α_ (N.tensorBimod P).X Y.X Q.X) ≫ (N.tensorBimod P).X ◁ Q.actLeft) :
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                              (coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
                                (Iso.hom (α_ (N.tensorBimod P).X Y.X Q.X) ≫ (N.tensorBimod P).X ◁ Q.actLeft)) ⟶
                            tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                              (coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
                                (Iso.hom (α_ (N.tensorBimod P).X Y.X Q.X) ≫ (N.tensorBimod P).X ◁ Q.actLeft)))
                        (Iso.hom
                              (α_ M.X W.X
                                (coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
                                  (Iso.hom (α_ (N.tensorBimod P).X Y.X Q.X) ≫ (N.tensorBimod P).X ◁ Q.actLeft))) ≫
                            M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q :
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                              (coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
                                (Iso.hom (α_ (N.tensorBimod P).X Y.X Q.X) ≫ (N.tensorBimod P).X ◁ Q.actLeft)) ⟶
                            tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                              (coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
                                (Iso.hom (α_ (N.tensorBimod P).X Y.X Q.X) ≫ (N.tensorBimod P).X ◁ Q.actLeft)))).obj
                    WalkingParallelPair.one ⟶
                  colimit
                    (parallelPair (X :=
                      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                        (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                          (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                      (Y :=
                      tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                        (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                          (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                      (M.actRight ▷
                          coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                            (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft) :
                        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                            (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                              (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ⟶
                          tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                            (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                              (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                      (Iso.hom
                            (α_ M.X W.X
                              (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft))) ≫
                          M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q) :
                        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                            (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                              (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ⟶
                          tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                            (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                              (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft))))) :
            tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                (tensorObj.{v₁, u₁} (𝒞 := inst)
                  (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)) Q.X) ⟶
              coequalizer (M.actRight ▷ (N.tensorBimod (P.tensorBimod Q)).X)
                (Iso.hom (α_ M.X W.X (N.tensorBimod (P.tensorBimod Q)).X) ≫
                  M.X ◁ (N.tensorBimod (P.tensorBimod Q)).actLeft)) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X) P.X) Q.X ⟶
      coequalizer (M.actRight ▷ (N.tensorBimod (P.tensorBimod Q)).X)
        (Iso.hom (α_ M.X W.X (N.tensorBimod (P.tensorBimod Q)).X) ≫ M.X ◁ (N.tensorBimod (P.tensorBimod Q)).actLeft))
  (coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
      Iso.hom (α_ (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft)) P.X Q.X) ≫
        coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft) ◁
            coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft) ≫
          CategoryStruct.comp (Y :=
            coequalizer (X :=
              tensorObj.{v₁, u₁} (𝒞 := inst)
                (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X) N.X)
                (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
              (M.actRight ▷ N.X ▷ coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))
              ((Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft) ▷
                coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
            (Iso.inv
                (PreservesCoequalizer.iso
                  (tensorRight (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                  (M.actRight ▷ N.X) (Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft)) :
              tensorObj.{v₁, u₁} (𝒞 := inst)
                  (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft))
                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
                coequalizer (X :=
                  tensorObj.{v₁, u₁} (𝒞 := inst)
                    (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X) N.X)
                    (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                  (M.actRight ▷ N.X ▷ coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))
                  ((Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft) ▷
                    coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
            (coequalizer.desc
                (Iso.hom (α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))) ≫
                  CategoryStruct.comp (Y :=
                    tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                      (coequalizer (X :=
                        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                          (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                        (Y :=
                        tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                          (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                        (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft) :
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
                            tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                        (Iso.hom
                              (α_ N.X X.X
                                (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))) ≫
                            N.X ◁ TensorBimod.actLeft P Q :
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
                            tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))))
                    (M.X ◁
                        coequalizer.π
                          (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))
                          (Iso.hom
                              (α_ N.X X.X
                                (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))) ≫
                            N.X ◁ TensorBimod.actLeft P Q) :
                      tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                          (tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                            (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))) ⟶
                        tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                          (coequalizer (X :=
                            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                            (Y :=
                            tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                            (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft) :
                              tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
                                tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                            (Iso.hom
                                  (α_ N.X X.X
                                    (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))) ≫
                                N.X ◁ TensorBimod.actLeft P Q :
                              tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
                                tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))))
                    (coequalizer.π
                        (M.actRight ▷
                          coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                            (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft))
                        (Iso.hom
                            (α_ M.X W.X
                              (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft))) ≫
                          M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)) :
                      tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                          (coequalizer (X :=
                            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                            (Y :=
                            tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                            (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft) :
                              tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
                                tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                            (Iso.hom
                                  (α_ N.X X.X
                                    (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))) ≫
                                N.X ◁ TensorBimod.actLeft P Q :
                              tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
                                tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))) ⟶
                        coequalizer (X :=
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                            (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                              (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                          (Y :=
                          tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                            (coequalizer (X :=
                              tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                                (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                              (Y :=
                              tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                                (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                              (N.actRight ▷
                                  coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft) :
                                tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                                    (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
                                  tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                                    (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                              (Iso.hom
                                    (α_ N.X X.X
                                      (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))) ≫
                                  N.X ◁ TensorBimod.actLeft P Q :
                                tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X)
                                    (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
                                  tensorObj.{v₁, u₁} (𝒞 := inst) N.X
                                    (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))))
                          (M.actRight ▷
                              coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft) :
                            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                                (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                  (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ⟶
                              tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                                (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                  (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))
                          (Iso.hom
                                (α_ M.X W.X
                                  (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                    (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft))) ≫
                              M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q) :
                            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                                (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                  (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)) ⟶
                              tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                                (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                                  (Iso.hom (α_ N.X X.X (P.tensorBimod Q).X) ≫ N.X ◁ (P.tensorBimod Q).actLeft)))))
                sorry :
              coequalizer (X :=
                  tensorObj.{v₁, u₁} (𝒞 := inst)
                    (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X) N.X)
                    (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                  (M.actRight ▷ N.X ▷ coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft))
                  ((Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft) ▷
                    coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)) ⟶
                coequalizer (M.actRight ▷ (N.tensorBimod (P.tensorBimod Q)).X)
                  (Iso.hom (α_ M.X W.X (N.tensorBimod (P.tensorBimod Q)).X) ≫
                    M.X ◁ (N.tensorBimod (P.tensorBimod Q)).actLeft)) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X) P.X) Q.X ⟶
      coequalizer (M.actRight ▷ (N.tensorBimod (P.tensorBimod Q)).X)
        (Iso.hom (α_ M.X W.X (N.tensorBimod (P.tensorBimod Q)).X) ≫ M.X ◁ (N.tensorBimod (P.tensorBimod Q)).actLeft))
Tactic: slice_lhs 5 6 => rw [ι_colimMap, parallelPairHom_app_one]