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
  

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.π (TensorBimod.actRight (tensorBimod M N) P ⊗ 𝟙 Q.X)
        ((α_ (TensorBimod.X (tensorBimod M N) P) Y.X Q.X).hom ≫ (𝟙 (TensorBimod.X (tensorBimod M N) P) ⊗ Q.actLeft)) ≫
      colimMap
          (parallelPairHom (TensorBimod.actRight (tensorBimod M N) P ⊗ 𝟙 Q.X)
            ((α_ (TensorBimod.X (tensorBimod M N) P) Y.X Q.X).hom ≫ (𝟙 (TensorBimod.X (tensorBimod M N) P) ⊗ Q.actLeft))
            (TensorBimod.actRight M (tensorBimod N P) ⊗ 𝟙 Q.X)
            ((α_ (TensorBimod.X M (tensorBimod N P)) Y.X Q.X).hom ≫ (𝟙 (TensorBimod.X M (tensorBimod N P)) ⊗ Q.actLeft))
            ((AssociatorBimod.hom M N P ⊗ 𝟙 Y.X) ⊗ 𝟙 Q.X) (AssociatorBimod.hom M N P ⊗ 𝟙 Q.X)
            (_ :
              ((tensorBimod (tensorBimod M N) P).actRight ⊗ 𝟙 Q.X) ≫
                  ((isoOfIso (Iso.mk (AssociatorBimod.hom M N P) (AssociatorBimod.inv M N P))
                          (_ :
                            (tensorBimod (tensorBimod M N) P).actLeft ≫ AssociatorBimod.hom M N P =
                              (𝟙 V.X ⊗ AssociatorBimod.hom M N P) ≫ (tensorBimod M (tensorBimod N P)).actLeft)
                          (_ :
                            (tensorBimod (tensorBimod M N) P).actRight ≫ AssociatorBimod.hom M N P =
                              (AssociatorBimod.hom M N P ⊗ 𝟙 Y.X) ≫
                                (tensorBimod M (tensorBimod N P)).actRight)).hom.hom ⊗
                    (𝟙 Q).hom) =
                (((isoOfIso (Iso.mk (AssociatorBimod.hom M N P) (AssociatorBimod.inv M N P))
                            (_ :
                              (tensorBimod (tensorBimod M N) P).actLeft ≫ AssociatorBimod.hom M N P =
                                (𝟙 V.X ⊗ AssociatorBimod.hom M N P) ≫ (tensorBimod M (tensorBimod N P)).actLeft)
                            (_ :
                              (tensorBimod (tensorBimod M N) P).actRight ≫ AssociatorBimod.hom M N P =
                                (AssociatorBimod.hom M N P ⊗ 𝟙 Y.X) ≫
                                  (tensorBimod M (tensorBimod N P)).actRight)).hom.hom ⊗
                      𝟙 Y.X) ⊗
                    (𝟙 Q).hom) ≫
                  ((tensorBimod M (tensorBimod N P)).actRight ⊗ 𝟙 Q.X))
            (_ :
              ((α_ (tensorBimod (tensorBimod M N) P).X Y.X Q.X).hom ≫
                    (𝟙 (tensorBimod (tensorBimod M N) P).X ⊗ Q.actLeft)) ≫
                  ((isoOfIso (Iso.mk (AssociatorBimod.hom M N P) (AssociatorBimod.inv M N P))
                          (_ :
                            (tensorBimod (tensorBimod M N) P).actLeft ≫ AssociatorBimod.hom M N P =
                              (𝟙 V.X ⊗ AssociatorBimod.hom M N P) ≫ (tensorBimod M (tensorBimod N P)).actLeft)
                          (_ :
                            (tensorBimod (tensorBimod M N) P).actRight ≫ AssociatorBimod.hom M N P =
                              (AssociatorBimod.hom M N P ⊗ 𝟙 Y.X) ≫
                                (tensorBimod M (tensorBimod N P)).actRight)).hom.hom ⊗
                    (𝟙 Q).hom) =
                (((isoOfIso (Iso.mk (AssociatorBimod.hom M N P) (AssociatorBimod.inv M N P))
                            (_ :
                              (tensorBimod (tensorBimod M N) P).actLeft ≫ AssociatorBimod.hom M N P =
                                (𝟙 V.X ⊗ AssociatorBimod.hom M N P) ≫ (tensorBimod M (tensorBimod N P)).actLeft)
                            (_ :
                              (tensorBimod (tensorBimod M N) P).actRight ≫ AssociatorBimod.hom M N P =
                                (AssociatorBimod.hom M N P ⊗ 𝟙 Y.X) ≫
                                  (tensorBimod M (tensorBimod N P)).actRight)).hom.hom ⊗
                      𝟙 Y.X) ⊗
                    (𝟙 Q).hom) ≫
                  (α_ (tensorBimod M (tensorBimod N P)).X Y.X Q.X).hom ≫
                    (𝟙 (tensorBimod M (tensorBimod N P)).X ⊗ Q.actLeft))) ≫
        AssociatorBimod.hom 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) ⊗ AssociatorBimod.hom N P Q) (𝟙 M.X ⊗ AssociatorBimod.hom N P Q)
              (_ :
                (M.actRight ⊗ 𝟙 (tensorBimod (tensorBimod N P) Q).X) ≫
                    ((𝟙 M).hom ⊗
                      (isoOfIso (Iso.mk (AssociatorBimod.hom 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 (AssociatorBimod.hom 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 (AssociatorBimod.hom 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 (AssociatorBimod.hom 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))) =
    coequalizer.π (TensorBimod.actRight (tensorBimod M N) P ⊗ 𝟙 Q.X)
        ((α_ (TensorBimod.X (tensorBimod M N) P) Y.X Q.X).hom ≫ (𝟙 (TensorBimod.X (tensorBimod M N) P) ⊗ Q.actLeft)) ≫
      AssociatorBimod.hom (tensorBimod M N) P Q ≫ AssociatorBimod.hom M N (tensorBimod P Q)

Next tactic:
dsimp only [AssociatorBimod.hom]