-- 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
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
⊢ (whiskerRight
          (isoOfIso
              { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯
              ⋯).hom
          Q ≫
        (isoOfIso
              { hom := AssociatorBimod.hom M (N.tensorBimod P) Q, inv := AssociatorBimod.inv M (N.tensorBimod P) Q,
                hom_inv_id := ⋯, inv_hom_id := ⋯ }
              ⋯ ⋯).hom ≫
          M.whiskerLeft
            (isoOfIso
                { hom := AssociatorBimod.hom N P Q, inv := AssociatorBimod.inv N P Q, hom_inv_id := ⋯, inv_hom_id := ⋯ }
                ⋯ ⋯).hom).hom =
    ((isoOfIso
            { hom := AssociatorBimod.hom (M.tensorBimod N) P Q, inv := AssociatorBimod.inv (M.tensorBimod N) P Q,
              hom_inv_id := ⋯, inv_hom_id := ⋯ }
            ⋯ ⋯).hom ≫
        (isoOfIso
            { hom := AssociatorBimod.hom M N (P.tensorBimod Q), inv := AssociatorBimod.inv M N (P.tensorBimod Q),
              hom_inv_id := ⋯, inv_hom_id := ⋯ }
            ⋯ ⋯).hom).hom

Tactic:
apply coequalizer.hom_ext

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 (((M.tensorBimod N).tensorBimod P).actRight ▷ Q.X)
        ((α_ ((M.tensorBimod N).tensorBimod P).X Y.X Q.X).hom ≫ ((M.tensorBimod N).tensorBimod P).X ◁ Q.actLeft) ≫
      (whiskerRight
            (isoOfIso
                { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ }
                ⋯ ⋯).hom
            Q ≫
          (isoOfIso
                { hom := AssociatorBimod.hom M (N.tensorBimod P) Q, inv := AssociatorBimod.inv M (N.tensorBimod P) Q,
                  hom_inv_id := ⋯, inv_hom_id := ⋯ }
                ⋯ ⋯).hom ≫
            M.whiskerLeft
              (isoOfIso
                  { hom := AssociatorBimod.hom N P Q, inv := AssociatorBimod.inv N P Q, hom_inv_id := ⋯,
                    inv_hom_id := ⋯ }
                  ⋯ ⋯).hom).hom =
    coequalizer.π (((M.tensorBimod N).tensorBimod P).actRight ▷ Q.X)
        ((α_ ((M.tensorBimod N).tensorBimod P).X Y.X Q.X).hom ≫ ((M.tensorBimod N).tensorBimod P).X ◁ Q.actLeft) ≫
      ((isoOfIso
              { hom := AssociatorBimod.hom (M.tensorBimod N) P Q, inv := AssociatorBimod.inv (M.tensorBimod N) P Q,
                hom_inv_id := ⋯, inv_hom_id := ⋯ }
              ⋯ ⋯).hom ≫
          (isoOfIso
              { hom := AssociatorBimod.hom M N (P.tensorBimod Q), inv := AssociatorBimod.inv M N (P.tensorBimod Q),
                hom_inv_id := ⋯, inv_hom_id := ⋯ }
              ⋯ ⋯).hom).hom