-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
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) :
    whiskerRight (associatorBimod M N P).hom Q ≫
      (associatorBimod M (N.tensorBimod P) Q).hom ≫
        whiskerLeft M (associatorBimod N P Q).hom =
      (associatorBimod (M.tensorBimod N) P Q).hom ≫
        (associatorBimod M N (P.tensorBimod Q)).hom

Proof state:
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
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.tensorBimod N) P ▷ Q.X)
        ((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
      colimMap
          (parallelPairHom (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
            ((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft)
            (TensorBimod.actRight M (N.tensorBimod P) ▷ Q.X)
            ((α_ (TensorBimod.X M (N.tensorBimod P)) Y.X Q.X).hom ≫ TensorBimod.X M (N.tensorBimod P) ◁ Q.actLeft)
            (AssociatorBimod.hom M N P ▷ Y.X ▷ Q.X) (AssociatorBimod.hom M N P ▷ Q.X) ⋯ ⋯) ≫
        AssociatorBimod.hom M (N.tensorBimod P) Q ≫
          colimMap
            (parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
              ((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
              (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
              ((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
              ((M.X ⊗ W.X) ◁ AssociatorBimod.hom N P Q) (M.X ◁ AssociatorBimod.hom N P Q) ⋯ ⋯) =
    coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
        ((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
      AssociatorBimod.hom (M.tensorBimod N) P Q ≫ AssociatorBimod.hom M N (P.tensorBimod Q)

Tactic:
dsimp only [AssociatorBimod.hom]

Result:
case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
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.tensorBimod N) P ▷ Q.X)
        ((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
      colimMap
          (parallelPairHom (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
            ((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft)
            (TensorBimod.actRight M (N.tensorBimod P) ▷ Q.X)
            ((α_ (TensorBimod.X M (N.tensorBimod P)) Y.X Q.X).hom ≫ TensorBimod.X M (N.tensorBimod P) ◁ Q.actLeft)
            (coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ▷ Y.X ▷ Q.X)
            (coequalizer.desc (AssociatorBimod.homAux M N P) ⋯ ▷ Q.X) ⋯ ⋯) ≫
        coequalizer.desc (AssociatorBimod.homAux M (N.tensorBimod P) Q) ⋯ ≫
          colimMap
            (parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
              ((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
              (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
              ((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
              ((M.X ⊗ W.X) ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯)
              (M.X ◁ coequalizer.desc (AssociatorBimod.homAux N P Q) ⋯) ⋯ ⋯) =
    coequalizer.π (TensorBimod.actRight (M.tensorBimod N) P ▷ Q.X)
        ((α_ (TensorBimod.X (M.tensorBimod N) P) Y.X Q.X).hom ≫ TensorBimod.X (M.tensorBimod N) P ◁ Q.actLeft) ≫
      coequalizer.desc (AssociatorBimod.homAux (M.tensorBimod N) P Q) ⋯ ≫
        coequalizer.desc (AssociatorBimod.homAux M N (P.tensorBimod Q)) ⋯