-- 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 :
  (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)))) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X N.X)
        (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))
| (CategoryStruct.comp (Y :=
      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.π (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))
      (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)))))
            ⋯ :
          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) M.X N.X)
        (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))

Tactic:
rw [π_tensor_id_preserves_coequalizer_inv_desc]

Premises used: π_tensor_id_preserves_coequalizer_inv_desc