-- 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
⊢ coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ▷ Q.X ≫
      AssociatorBimod.homAux M (N.tensorBimod P) 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 (AssociatorBimod.homAux N P Q) ⋯)
            (M.X ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯) ⋯ ⋯) =
    AssociatorBimod.homAux (M.tensorBimod N) P Q ≫ coequalizer.desc (AssociatorBimod.homAux M N (P.tensorBimod Q)) ⋯
have h : 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)))
              sorry))
          sorry ▷
        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)))
              sorry) :
          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)))
                      sorry))
                  sorry)
              (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)))
                      sorry))
                  sorry)
              sorry sorry) :
          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)))
            sorry) :
        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))))
              sorry))
          sorry :
        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: dsimp [AssociatorBimod.homAux]