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]
  

Proof state:
case h.h
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 (AssociatorBimod.homAux M N P)
          (_ :
            ((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) ≫
      AssociatorBimod.homAux M (tensorBimod N P) Q ≫
        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 (AssociatorBimod.homAux N P Q)
                (_ :
                  ((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 (AssociatorBimod.homAux N P Q)
                (_ :
                  ((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 (AssociatorBimod.homAux N P Q)
                              (_ :
                                ((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 (AssociatorBimod.homAux N P Q)
                              (_ :
                                ((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 (AssociatorBimod.homAux N P Q)
                              (_ :
                                ((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 (AssociatorBimod.homAux N P Q)
                              (_ :
                                ((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))) =
    AssociatorBimod.homAux (tensorBimod M N) P Q ≫
      coequalizer.desc (AssociatorBimod.homAux M N (tensorBimod P Q))
        (_ :
          ((tensorBimod M N).actRight ⊗ 𝟙 (tensorBimod P Q).X) ≫ AssociatorBimod.homAux M N (tensorBimod P Q) =
            ((α_ (tensorBimod M N).X X.X (tensorBimod P Q).X).hom ≫
                (𝟙 (tensorBimod M N).X ⊗ (tensorBimod P Q).actLeft)) ≫
              AssociatorBimod.homAux M N (tensorBimod P Q))

Next tactic:
dsimp [AssociatorBimod.homAux]