-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
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
a :
  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)))))
              ⋯))
          ⋯ :
      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)))))
| (CategoryStruct.comp (Y :=
      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 ▷
            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) 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)) ⟶
          coequalizer (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))))
      (colimMap
          (parallelPairHom
            (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))
            (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)
            (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 W.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)))))
                    ⋯))
                ⋯)
            (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)))))
                    ⋯))
                ⋯)
            ⋯ ⋯) :
        colimit
            (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 ((N.tensorBimod P).actRight ▷ Q.X)
                  (Iso.hom (α_ (N.tensorBimod P).X Y.X Q.X) ≫ (N.tensorBimod P).X ◁ 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)))) ⟶
          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
        (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)) ⟶
      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:
rw [ι_colimMap, parallelPairHom_app_one]

Premises used: CategoryTheory.Limits.ι_colimMap, CategoryTheory.Limits.parallelPairHom_app_one