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 => 

Proof state:
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.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))
        ((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 Y.X) ⊗
          𝟙 Q.X)
        (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)
        (_ :
          ((tensorBimod (tensorBimod M N) P).actRight ⊗ 𝟙 Q.X) ≫
              ((isoOfIso
                      (Iso.mk
                        (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))
                        (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
                          (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))
                          (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
                        (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))
                        (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
                          (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))
                          (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)))
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 (AssociatorBimod.homAux M (tensorBimod N P) Q)
    (_ :
      ((tensorBimod M (tensorBimod N P)).actRight ⊗ 𝟙 Q.X) ≫ AssociatorBimod.homAux M (tensorBimod N P) Q =
        ((α_ (tensorBimod M (tensorBimod N P)).X Y.X Q.X).hom ≫ (𝟙 (tensorBimod M (tensorBimod N P)).X ⊗ Q.actLeft)) ≫
          AssociatorBimod.homAux M (tensorBimod N P) Q)
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 (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)))

Next tactic:
rw [ι_colimMap, parallelPairHom_app_one]