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
  

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 (tensorBimod M 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)) ≫
      (Hom.mk
            (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)))) ≫
          (isoOfIso (Iso.mk (AssociatorBimod.hom M (tensorBimod N P) Q) (AssociatorBimod.inv M (tensorBimod N P) Q))
                (_ :
                  (tensorBimod (tensorBimod M (tensorBimod N P)) Q).actLeft ≫
                      AssociatorBimod.hom M (tensorBimod N P) Q =
                    (𝟙 V.X ⊗ AssociatorBimod.hom M (tensorBimod N P) Q) ≫
                      (tensorBimod M (tensorBimod (tensorBimod N P) Q)).actLeft)
                (_ :
                  (tensorBimod (tensorBimod M (tensorBimod N P)) Q).actRight ≫
                      AssociatorBimod.hom M (tensorBimod N P) Q =
                    (AssociatorBimod.hom M (tensorBimod N P) Q ⊗ 𝟙 Z.X) ≫
                      (tensorBimod M (tensorBimod (tensorBimod N P) Q)).actRight)).hom ≫
            Hom.mk
              (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))))).hom =
    coequalizer.π ((tensorBimod (tensorBimod M 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 (tensorBimod M N) P Q) (AssociatorBimod.inv (tensorBimod M N) P Q))
              (_ :
                (tensorBimod (tensorBimod (tensorBimod M N) P) Q).actLeft ≫ AssociatorBimod.hom (tensorBimod M N) P Q =
                  (𝟙 V.X ⊗ AssociatorBimod.hom (tensorBimod M N) P Q) ≫
                    (tensorBimod (tensorBimod M N) (tensorBimod P Q)).actLeft)
              (_ :
                (tensorBimod (tensorBimod (tensorBimod M N) P) Q).actRight ≫ AssociatorBimod.hom (tensorBimod M N) P Q =
                  (AssociatorBimod.hom (tensorBimod M N) P Q ⊗ 𝟙 Z.X) ≫
                    (tensorBimod (tensorBimod M N) (tensorBimod P Q)).actRight)).hom ≫
          (isoOfIso (Iso.mk (AssociatorBimod.hom M N (tensorBimod P Q)) (AssociatorBimod.inv M N (tensorBimod P Q)))
              (_ :
                (tensorBimod (tensorBimod M N) (tensorBimod P Q)).actLeft ≫ AssociatorBimod.hom M N (tensorBimod P Q) =
                  (𝟙 V.X ⊗ AssociatorBimod.hom M N (tensorBimod P Q)) ≫
                    (tensorBimod M (tensorBimod N (tensorBimod P Q))).actLeft)
              (_ :
                (tensorBimod (tensorBimod M N) (tensorBimod P Q)).actRight ≫ AssociatorBimod.hom M N (tensorBimod P Q) =
                  (AssociatorBimod.hom M N (tensorBimod P Q) ⊗ 𝟙 Z.X) ≫
                    (tensorBimod M (tensorBimod N (tensorBimod P Q))).actRight)).hom).hom

Next tactic:
dsimp