-- 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
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X (M.tensorBimod N) P) Q.X ⟶
      TensorBimod.X M (N.tensorBimod (P.tensorBimod Q)))
    (coequalizer.desc
            (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) P.X)
                (M.actRight ▷ N.X ▷ P.X) ((Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft) ▷ P.X))
              (Iso.inv
                (PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X)
                  (Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft)))
              (coequalizer.desc
                (Iso.hom (α_ M.X N.X P.X) ≫
                  CategoryStruct.comp (Y :=
                    tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                      (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)))
                    (M.X ◁ coequalizer.π (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft) :
                      tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X) ⟶
                        tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                          (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)))
                    (coequalizer.π (M.actRight ▷ TensorBimod.X N P)
                        (Iso.hom (α_ M.X W.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P) :
                      tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                          (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X X.X P.X) ≫ N.X ◁ P.actLeft)) ⟶
                        coequalizer (X :=
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X) (TensorBimod.X N P))
                          (M.actRight ▷ TensorBimod.X N P)
                          (Iso.hom (α_ M.X W.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P)))
                ⋯))
            ⋯ ▷
          Q.X ≫
        CategoryStruct.comp (Y := TensorBimod.X M ((N.tensorBimod P).tensorBimod Q))
          (CategoryStruct.comp (Y :=
              coequalizer (X :=
                tensorObj.{v₁, u₁} (𝒞 := inst)
                  (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X) (TensorBimod.X N P)) Q.X)
                (M.actRight ▷ TensorBimod.X N P ▷ Q.X)
                ((Iso.hom (α_ M.X W.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P) ▷ Q.X))
              (Iso.inv
                (PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P)
                  (Iso.hom (α_ M.X W.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P)))
              (coequalizer.desc
                (Iso.hom (α_ M.X (TensorBimod.X N P) Q.X) ≫
                  CategoryStruct.comp (Y :=
                    tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                      (coequalizer (TensorBimod.actRight N P ▷ Q.X)
                        (Iso.hom (α_ (TensorBimod.X N P) Y.X Q.X) ≫ TensorBimod.X N P ◁ Q.actLeft)))
                    (M.X ◁
                        coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
                          (Iso.hom (α_ (TensorBimod.X N P) Y.X Q.X) ≫ TensorBimod.X N P ◁ Q.actLeft) :
                      tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X N P) Q.X) ⟶
                        tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                          (coequalizer (TensorBimod.actRight N P ▷ Q.X)
                            (Iso.hom (α_ (TensorBimod.X N P) Y.X Q.X) ≫ TensorBimod.X N P ◁ Q.actLeft)))
                    (coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
                        (Iso.hom (α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)) ≫
                          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_ (TensorBimod.X N P) Y.X Q.X) ≫ TensorBimod.X N P ◁ Q.actLeft)) ⟶
                        coequalizer (X :=
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                            (TensorBimod.X (N.tensorBimod P) Q))
                          (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
                          (Iso.hom (α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)) ≫
                            M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)))
                ⋯) :
            tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M (N.tensorBimod P)) Q.X ⟶
              TensorBimod.X M ((N.tensorBimod P).tensorBimod Q))
          (colimMap
              (parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
                (Iso.hom (α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)) ≫
                  M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
                (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
                (Iso.hom (α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))) ≫
                  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 ▷ TensorBimod.X P Q)
                                (Iso.hom (α_ N.X X.X (TensorBimod.X P Q)) ≫ 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)
                                    (TensorBimod.X P Q))
                                  (N.actRight ▷ TensorBimod.X P Q)
                                  (Iso.hom (α_ N.X X.X (TensorBimod.X P Q)) ≫ N.X ◁ TensorBimod.actLeft P Q)))
                        ⋯))
                    ⋯)
                (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 ▷ TensorBimod.X P Q)
                                (Iso.hom (α_ N.X X.X (TensorBimod.X P Q)) ≫ 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)
                                    (TensorBimod.X P Q))
                                  (N.actRight ▷ TensorBimod.X P Q)
                                  (Iso.hom (α_ N.X X.X (TensorBimod.X P Q)) ≫ N.X ◁ TensorBimod.actLeft P Q)))
                        ⋯))
                    ⋯)
                ⋯ ⋯) :
            colimit
                (parallelPair (X :=
                  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                    (TensorBimod.X (N.tensorBimod P) Q))
                  (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
                  (Iso.hom (α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)) ≫
                    M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)) ⟶
              colimit
                (parallelPair (X :=
                  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                    (TensorBimod.X N (P.tensorBimod Q)))
                  (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
                  (Iso.hom (α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))) ≫
                    M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X (M.tensorBimod N) P) Q.X ⟶
        TensorBimod.X M (N.tensorBimod (P.tensorBimod Q)))
    (CategoryStruct.comp (Y := TensorBimod.X (M.tensorBimod N) (P.tensorBimod Q))
        (CategoryStruct.comp (Y :=
            coequalizer (X :=
              tensorObj.{v₁, u₁} (𝒞 := inst)
                (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) X.X) P.X) Q.X)
              (TensorBimod.actRight M N ▷ P.X ▷ Q.X)
              ((Iso.hom (α_ (TensorBimod.X M N) X.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft) ▷ Q.X))
            (Iso.inv
              (PreservesCoequalizer.iso (tensorRight Q.X) (TensorBimod.actRight M N ▷ P.X)
                (Iso.hom (α_ (TensorBimod.X M N) X.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft)))
            (coequalizer.desc
              (Iso.hom (α_ (TensorBimod.X M N) P.X Q.X) ≫
                CategoryStruct.comp (Y :=
                  tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N)
                    (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                  (TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft) :
                    tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) ⟶
                      tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N)
                        (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X Y.X Q.X) ≫ P.X ◁ Q.actLeft)))
                  (coequalizer.π (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
                      (Iso.hom (α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)) ≫
                        TensorBimod.X M N ◁ TensorBimod.actLeft P Q) :
                    tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N)
                        (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) (TensorBimod.X M N) X.X)
                          (TensorBimod.X P Q))
                        (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
                        (Iso.hom (α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)) ≫
                          TensorBimod.X M N ◁ TensorBimod.actLeft P Q)))
              ⋯) :
          tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X (M.tensorBimod N) P) Q.X ⟶
            TensorBimod.X (M.tensorBimod N) (P.tensorBimod Q))
        (coequalizer.desc
            (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) (TensorBimod.X P Q))
                (M.actRight ▷ N.X ▷ TensorBimod.X P Q)
                ((Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft) ▷ TensorBimod.X P Q))
              (Iso.inv
                (PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
                  (Iso.hom (α_ M.X W.X N.X) ≫ M.X ◁ N.actLeft)))
              (coequalizer.desc
                (Iso.hom (α_ M.X N.X (TensorBimod.X P Q)) ≫
                  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) (TensorBimod.X P Q))
                        (N.actRight ▷ TensorBimod.X P Q)
                        (Iso.hom (α_ N.X X.X (TensorBimod.X P Q)) ≫ N.X ◁ TensorBimod.actLeft P Q)))
                    (M.X ◁
                        coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
                          (Iso.hom (α_ N.X X.X (TensorBimod.X P Q)) ≫ N.X ◁ TensorBimod.actLeft P Q) :
                      tensorObj.{v₁, u₁} (𝒞 := inst) M.X (tensorObj.{v₁, u₁} (𝒞 := inst) N.X (TensorBimod.X P Q)) ⟶
                        tensorObj.{v₁, u₁} (𝒞 := inst) M.X
                          (coequalizer (X :=
                            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X X.X) (TensorBimod.X P Q))
                            (N.actRight ▷ TensorBimod.X P Q)
                            (Iso.hom (α_ N.X X.X (TensorBimod.X P Q)) ≫ N.X ◁ TensorBimod.actLeft P Q)))
                    (coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
                        (Iso.hom (α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))) ≫
                          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) (TensorBimod.X P Q))
                            (N.actRight ▷ TensorBimod.X P Q)
                            (Iso.hom (α_ N.X X.X (TensorBimod.X P Q)) ≫ N.X ◁ TensorBimod.actLeft P Q)) ⟶
                        coequalizer (X :=
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X W.X)
                            (TensorBimod.X N (P.tensorBimod Q)))
                          (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
                          (Iso.hom (α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))) ≫
                            M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))))
                ⋯))
            ⋯ :
          coequalizer (X :=
              tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X M N) X.X)
                (TensorBimod.X P Q))
              (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
              (Iso.hom (α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)) ≫ TensorBimod.X M N ◁ TensorBimod.actLeft P Q) ⟶
            TensorBimod.X M (N.tensorBimod (P.tensorBimod Q))) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X (M.tensorBimod N) P) Q.X ⟶
        TensorBimod.X M (N.tensorBimod (P.tensorBimod Q)))

Tactic:
(cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 ?_

Premises used: CategoryTheory.Limits.colimit, CategoryTheory.Functor.map, CategoryTheory.MonoidalCategory.tensorRight, Bimod.TensorBimod.X, Iff.mp, CategoryTheory.Limits.coequalizer.π, CategoryTheory.cancel_epi