theorem pentagon_bimod {V W X Y Z : Mon_ C} (M : Bimod V W) (N : Bimod W X) (P : Bimod X Y)
    (Q : Bimod Y Z) :
    tensorHom (associatorBimod M N P).hom (𝟙 Q) ≫
      (associatorBimod M (N.tensorBimod P) Q).hom ≫
        tensorHom (𝟙 M) (associatorBimod N P Q).hom =
      (associatorBimod (M.tensorBimod N) P Q).hom ≫
        (associatorBimod M N (P.tensorBimod Q)).hom := by
  dsimp [tensorHom, associatorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  dsimp only [AssociatorBimod.hom]
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  slice_lhs 2 3 => rw [coequalizer.π_desc]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [AssociatorBimod.homAux]
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp
  slice_lhs 1 2 => 

Proof state:
case a.a.a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
| (coequalizer.π (TensorBimod.actRight M N ⊗ 𝟙 P.X)
        ((α_ (TensorBimod.X M N) X.X P.X).hom ≫ (𝟙 (TensorBimod.X M N) ⊗ P.actLeft)) ⊗
      𝟙 Q.X) ≫
    (coequalizer.desc
        ((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ⊗ 𝟙 N.X)
              ((α_ M.X W.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft))).inv ≫
          coequalizer.desc
            ((α_ M.X N.X P.X).hom ≫
              (𝟙 M.X ⊗ coequalizer.π (N.actRight ⊗ 𝟙 P.X) ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft))) ≫
                coequalizer.π (M.actRight ⊗ 𝟙 (TensorBimod.X N P))
                  ((α_ M.X W.X (TensorBimod.X N P)).hom ≫ (𝟙 M.X ⊗ TensorBimod.actLeft N P)))
            (_ :
              (tensorRight P.X).map (M.actRight ⊗ 𝟙 N.X) ≫
                  (α_ M.X N.X P.X).hom ≫
                    (𝟙 M.X ⊗ coequalizer.π (N.actRight ⊗ 𝟙 P.X) ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft))) ≫
                      coequalizer.π (M.actRight ⊗ 𝟙 (tensorBimod N P).X)
                        ((α_ M.X W.X (tensorBimod N P).X).hom ≫ (𝟙 M.X ⊗ (tensorBimod N P).actLeft)) =
                (tensorRight P.X).map ((α_ M.X W.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft)) ≫
                  (α_ M.X N.X P.X).hom ≫
                    (𝟙 M.X ⊗ coequalizer.π (N.actRight ⊗ 𝟙 P.X) ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft))) ≫
                      coequalizer.π (M.actRight ⊗ 𝟙 (tensorBimod N P).X)
                        ((α_ M.X W.X (tensorBimod N P).X).hom ≫ (𝟙 M.X ⊗ (tensorBimod N P).actLeft))))
        (_ :
          ((tensorBimod M N).actRight ⊗ 𝟙 P.X) ≫ AssociatorBimod.homAux M N P =
            ((α_ (tensorBimod M N).X X.X P.X).hom ≫ (𝟙 (tensorBimod M N).X ⊗ P.actLeft)) ≫
              AssociatorBimod.homAux M N P) ⊗
      𝟙 Q.X)
case a.a.a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
| (PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ⊗ 𝟙 (TensorBimod.X N P))
      ((α_ M.X W.X (TensorBimod.X N P)).hom ≫ (𝟙 M.X ⊗ TensorBimod.actLeft N P))).inv
case a.a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
| coequalizer.desc
    ((α_ M.X (TensorBimod.X N P) Q.X).hom ≫
      (𝟙 M.X ⊗
          coequalizer.π (TensorBimod.actRight N P ⊗ 𝟙 Q.X)
            ((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ (𝟙 (TensorBimod.X N P) ⊗ Q.actLeft))) ≫
        coequalizer.π (M.actRight ⊗ 𝟙 (TensorBimod.X (tensorBimod N P) Q))
          ((α_ M.X W.X (TensorBimod.X (tensorBimod N P) Q)).hom ≫ (𝟙 M.X ⊗ TensorBimod.actLeft (tensorBimod N P) Q)))
    (_ :
      (tensorRight Q.X).map (M.actRight ⊗ 𝟙 (tensorBimod N P).X) ≫
          (α_ M.X (tensorBimod N P).X Q.X).hom ≫
            (𝟙 M.X ⊗
                coequalizer.π ((tensorBimod N P).actRight ⊗ 𝟙 Q.X)
                  ((α_ (tensorBimod N P).X Y.X Q.X).hom ≫ (𝟙 (tensorBimod N P).X ⊗ Q.actLeft))) ≫
              coequalizer.π (M.actRight ⊗ 𝟙 (tensorBimod (tensorBimod N P) Q).X)
                ((α_ M.X W.X (tensorBimod (tensorBimod N P) Q).X).hom ≫
                  (𝟙 M.X ⊗ (tensorBimod (tensorBimod N P) Q).actLeft)) =
        (tensorRight Q.X).map ((α_ M.X W.X (tensorBimod N P).X).hom ≫ (𝟙 M.X ⊗ (tensorBimod N P).actLeft)) ≫
          (α_ M.X (tensorBimod N P).X Q.X).hom ≫
            (𝟙 M.X ⊗
                coequalizer.π ((tensorBimod N P).actRight ⊗ 𝟙 Q.X)
                  ((α_ (tensorBimod N P).X Y.X Q.X).hom ≫ (𝟙 (tensorBimod N P).X ⊗ Q.actLeft))) ≫
              coequalizer.π (M.actRight ⊗ 𝟙 (tensorBimod (tensorBimod N P) Q).X)
                ((α_ M.X W.X (tensorBimod (tensorBimod N P) Q).X).hom ≫
                  (𝟙 M.X ⊗ (tensorBimod (tensorBimod N P) Q).actLeft)))
case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
V W X Y Z : Mon_ C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
| colimMap
    (parallelPairHom (M.actRight ⊗ 𝟙 (TensorBimod.X (tensorBimod N P) Q))
      ((α_ M.X W.X (TensorBimod.X (tensorBimod N P) Q)).hom ≫ (𝟙 M.X ⊗ TensorBimod.actLeft (tensorBimod N P) Q))
      (M.actRight ⊗ 𝟙 (TensorBimod.X N (tensorBimod P Q)))
      ((α_ M.X W.X (TensorBimod.X N (tensorBimod P Q))).hom ≫ (𝟙 M.X ⊗ TensorBimod.actLeft N (tensorBimod P Q)))
      ((𝟙 M.X ⊗ 𝟙 W.X) ⊗
        coequalizer.desc
          ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ⊗ 𝟙 P.X)
                ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft))).inv ≫
            coequalizer.desc
              ((α_ N.X P.X Q.X).hom ≫
                (𝟙 N.X ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                  coequalizer.π (N.actRight ⊗ 𝟙 (TensorBimod.X P Q))
                    ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ (𝟙 N.X ⊗ TensorBimod.actLeft P Q)))
              (_ :
                (tensorRight Q.X).map (N.actRight ⊗ 𝟙 P.X) ≫
                    (α_ N.X P.X Q.X).hom ≫
                      (𝟙 N.X ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                        coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                          ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft)) =
                  (tensorRight Q.X).map ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft)) ≫
                    (α_ N.X P.X Q.X).hom ≫
                      (𝟙 N.X ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                        coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                          ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft))))
          (_ :
            ((tensorBimod N P).actRight ⊗ 𝟙 Q.X) ≫ AssociatorBimod.homAux N P Q =
              ((α_ (tensorBimod N P).X Y.X Q.X).hom ≫ (𝟙 (tensorBimod N P).X ⊗ Q.actLeft)) ≫
                AssociatorBimod.homAux N P Q))
      (𝟙 M.X ⊗
        coequalizer.desc
          ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ⊗ 𝟙 P.X)
                ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft))).inv ≫
            coequalizer.desc
              ((α_ N.X P.X Q.X).hom ≫
                (𝟙 N.X ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                  coequalizer.π (N.actRight ⊗ 𝟙 (TensorBimod.X P Q))
                    ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ (𝟙 N.X ⊗ TensorBimod.actLeft P Q)))
              (_ :
                (tensorRight Q.X).map (N.actRight ⊗ 𝟙 P.X) ≫
                    (α_ N.X P.X Q.X).hom ≫
                      (𝟙 N.X ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                        coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                          ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft)) =
                  (tensorRight Q.X).map ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft)) ≫
                    (α_ N.X P.X Q.X).hom ≫
                      (𝟙 N.X ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                        coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                          ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft))))
          (_ :
            ((tensorBimod N P).actRight ⊗ 𝟙 Q.X) ≫ AssociatorBimod.homAux N P Q =
              ((α_ (tensorBimod N P).X Y.X Q.X).hom ≫ (𝟙 (tensorBimod N P).X ⊗ Q.actLeft)) ≫
                AssociatorBimod.homAux N P Q))
      (_ :
        (M.actRight ⊗ 𝟙 (tensorBimod (tensorBimod N P) Q).X) ≫
            ((𝟙 M).hom ⊗
              (isoOfIso
                    (Iso.mk
                      (coequalizer.desc
                        ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ⊗ 𝟙 P.X)
                              ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft))).inv ≫
                          coequalizer.desc
                            ((α_ N.X P.X Q.X).hom ≫
                              (𝟙 N.X ⊗
                                  coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                coequalizer.π (N.actRight ⊗ 𝟙 (TensorBimod.X P Q))
                                  ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ (𝟙 N.X ⊗ TensorBimod.actLeft P Q)))
                            (_ :
                              (tensorRight Q.X).map (N.actRight ⊗ 𝟙 P.X) ≫
                                  (α_ N.X P.X Q.X).hom ≫
                                    (𝟙 N.X ⊗
                                        coequalizer.π (P.actRight ⊗ 𝟙 Q.X)
                                          ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                      coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                                        ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft)) =
                                (tensorRight Q.X).map ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft)) ≫
                                  (α_ N.X P.X Q.X).hom ≫
                                    (𝟙 N.X ⊗
                                        coequalizer.π (P.actRight ⊗ 𝟙 Q.X)
                                          ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                      coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                                        ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft))))
                        (_ :
                          ((tensorBimod N P).actRight ⊗ 𝟙 Q.X) ≫ AssociatorBimod.homAux N P Q =
                            ((α_ (tensorBimod N P).X Y.X Q.X).hom ≫ (𝟙 (tensorBimod N P).X ⊗ Q.actLeft)) ≫
                              AssociatorBimod.homAux N P Q))
                      (AssociatorBimod.inv N P Q))
                    (_ :
                      (tensorBimod (tensorBimod N P) Q).actLeft ≫ AssociatorBimod.hom N P Q =
                        (𝟙 W.X ⊗ AssociatorBimod.hom N P Q) ≫ (tensorBimod N (tensorBimod P Q)).actLeft)
                    (_ :
                      (tensorBimod (tensorBimod N P) Q).actRight ≫ AssociatorBimod.hom N P Q =
                        (AssociatorBimod.hom N P Q ⊗ 𝟙 Z.X) ≫ (tensorBimod N (tensorBimod P Q)).actRight)).hom.hom) =
          (((𝟙 M).hom ⊗ 𝟙 W.X) ⊗
              (isoOfIso
                    (Iso.mk
                      (coequalizer.desc
                        ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ⊗ 𝟙 P.X)
                              ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft))).inv ≫
                          coequalizer.desc
                            ((α_ N.X P.X Q.X).hom ≫
                              (𝟙 N.X ⊗
                                  coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                coequalizer.π (N.actRight ⊗ 𝟙 (TensorBimod.X P Q))
                                  ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ (𝟙 N.X ⊗ TensorBimod.actLeft P Q)))
                            (_ :
                              (tensorRight Q.X).map (N.actRight ⊗ 𝟙 P.X) ≫
                                  (α_ N.X P.X Q.X).hom ≫
                                    (𝟙 N.X ⊗
                                        coequalizer.π (P.actRight ⊗ 𝟙 Q.X)
                                          ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                      coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                                        ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft)) =
                                (tensorRight Q.X).map ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft)) ≫
                                  (α_ N.X P.X Q.X).hom ≫
                                    (𝟙 N.X ⊗
                                        coequalizer.π (P.actRight ⊗ 𝟙 Q.X)
                                          ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                      coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                                        ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft))))
                        (_ :
                          ((tensorBimod N P).actRight ⊗ 𝟙 Q.X) ≫ AssociatorBimod.homAux N P Q =
                            ((α_ (tensorBimod N P).X Y.X Q.X).hom ≫ (𝟙 (tensorBimod N P).X ⊗ Q.actLeft)) ≫
                              AssociatorBimod.homAux N P Q))
                      (AssociatorBimod.inv N P Q))
                    (_ :
                      (tensorBimod (tensorBimod N P) Q).actLeft ≫ AssociatorBimod.hom N P Q =
                        (𝟙 W.X ⊗ AssociatorBimod.hom N P Q) ≫ (tensorBimod N (tensorBimod P Q)).actLeft)
                    (_ :
                      (tensorBimod (tensorBimod N P) Q).actRight ≫ AssociatorBimod.hom N P Q =
                        (AssociatorBimod.hom N P Q ⊗ 𝟙 Z.X) ≫ (tensorBimod N (tensorBimod P Q)).actRight)).hom.hom) ≫
            (M.actRight ⊗ 𝟙 (tensorBimod N (tensorBimod P Q)).X))
      (_ :
        ((α_ M.X W.X (tensorBimod (tensorBimod N P) Q).X).hom ≫ (𝟙 M.X ⊗ (tensorBimod (tensorBimod N P) Q).actLeft)) ≫
            ((𝟙 M).hom ⊗
              (isoOfIso
                    (Iso.mk
                      (coequalizer.desc
                        ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ⊗ 𝟙 P.X)
                              ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft))).inv ≫
                          coequalizer.desc
                            ((α_ N.X P.X Q.X).hom ≫
                              (𝟙 N.X ⊗
                                  coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                coequalizer.π (N.actRight ⊗ 𝟙 (TensorBimod.X P Q))
                                  ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ (𝟙 N.X ⊗ TensorBimod.actLeft P Q)))
                            (_ :
                              (tensorRight Q.X).map (N.actRight ⊗ 𝟙 P.X) ≫
                                  (α_ N.X P.X Q.X).hom ≫
                                    (𝟙 N.X ⊗
                                        coequalizer.π (P.actRight ⊗ 𝟙 Q.X)
                                          ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                      coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                                        ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft)) =
                                (tensorRight Q.X).map ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft)) ≫
                                  (α_ N.X P.X Q.X).hom ≫
                                    (𝟙 N.X ⊗
                                        coequalizer.π (P.actRight ⊗ 𝟙 Q.X)
                                          ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                      coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                                        ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft))))
                        (_ :
                          ((tensorBimod N P).actRight ⊗ 𝟙 Q.X) ≫ AssociatorBimod.homAux N P Q =
                            ((α_ (tensorBimod N P).X Y.X Q.X).hom ≫ (𝟙 (tensorBimod N P).X ⊗ Q.actLeft)) ≫
                              AssociatorBimod.homAux N P Q))
                      (AssociatorBimod.inv N P Q))
                    (_ :
                      (tensorBimod (tensorBimod N P) Q).actLeft ≫ AssociatorBimod.hom N P Q =
                        (𝟙 W.X ⊗ AssociatorBimod.hom N P Q) ≫ (tensorBimod N (tensorBimod P Q)).actLeft)
                    (_ :
                      (tensorBimod (tensorBimod N P) Q).actRight ≫ AssociatorBimod.hom N P Q =
                        (AssociatorBimod.hom N P Q ⊗ 𝟙 Z.X) ≫ (tensorBimod N (tensorBimod P Q)).actRight)).hom.hom) =
          (((𝟙 M).hom ⊗ 𝟙 W.X) ⊗
              (isoOfIso
                    (Iso.mk
                      (coequalizer.desc
                        ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ⊗ 𝟙 P.X)
                              ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft))).inv ≫
                          coequalizer.desc
                            ((α_ N.X P.X Q.X).hom ≫
                              (𝟙 N.X ⊗
                                  coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                coequalizer.π (N.actRight ⊗ 𝟙 (TensorBimod.X P Q))
                                  ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ (𝟙 N.X ⊗ TensorBimod.actLeft P Q)))
                            (_ :
                              (tensorRight Q.X).map (N.actRight ⊗ 𝟙 P.X) ≫
                                  (α_ N.X P.X Q.X).hom ≫
                                    (𝟙 N.X ⊗
                                        coequalizer.π (P.actRight ⊗ 𝟙 Q.X)
                                          ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                      coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                                        ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft)) =
                                (tensorRight Q.X).map ((α_ N.X X.X P.X).hom ≫ (𝟙 N.X ⊗ P.actLeft)) ≫
                                  (α_ N.X P.X Q.X).hom ≫
                                    (𝟙 N.X ⊗
                                        coequalizer.π (P.actRight ⊗ 𝟙 Q.X)
                                          ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
                                      coequalizer.π (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
                                        ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft))))
                        (_ :
                          ((tensorBimod N P).actRight ⊗ 𝟙 Q.X) ≫ AssociatorBimod.homAux N P Q =
                            ((α_ (tensorBimod N P).X Y.X Q.X).hom ≫ (𝟙 (tensorBimod N P).X ⊗ Q.actLeft)) ≫
                              AssociatorBimod.homAux N P Q))
                      (AssociatorBimod.inv N P Q))
                    (_ :
                      (tensorBimod (tensorBimod N P) Q).actLeft ≫ AssociatorBimod.hom N P Q =
                        (𝟙 W.X ⊗ AssociatorBimod.hom N P Q) ≫ (tensorBimod N (tensorBimod P Q)).actLeft)
                    (_ :
                      (tensorBimod (tensorBimod N P) Q).actRight ≫ AssociatorBimod.hom N P Q =
                        (AssociatorBimod.hom N P Q ⊗ 𝟙 Z.X) ≫ (tensorBimod N (tensorBimod P Q)).actRight)).hom.hom) ≫
            (α_ M.X W.X (tensorBimod N (tensorBimod P Q)).X).hom ≫ (𝟙 M.X ⊗ (tensorBimod N (tensorBimod P Q)).actLeft)))

Next tactic:
rw [← comp_tensor_id, coequalizer.π_desc]