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 => rw [ι_colimMap, parallelPairHom_app_one]
  slice_lhs 2 3 => rw [coequalizer.π_desc]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [AssociatorBimod.homAux]
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp
  slice_lhs 1 2 => rw [← comp_tensor_id, coequalizer.π_desc]
  slice_rhs 1 3 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_rhs 3 4 => rw [coequalizer.π_desc]
  refine' (cancel_epi ((tensorRight _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp
  slice_lhs 1 2 =>
    rw [← comp_tensor_id, π_tensor_id_preserves_coequalizer_inv_desc, comp_tensor_id,
      comp_tensor_id]
  slice_lhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  dsimp only [TensorBimod.X]
  slice_lhs 2 3 => rw [associator_naturality]
  slice_lhs 5 6 => rw [ι_colimMap, parallelPairHom_app_one]
  slice_lhs 4 5 => rw [← id_tensor_comp, coequalizer.π_desc]
  slice_lhs 3 4 =>
    rw [← id_tensor_comp, π_tensor_id_preserves_coequalizer_inv_desc, id_tensor_comp,
      id_tensor_comp]
  slice_rhs 1 2 => rw [associator_naturality]
  slice_rhs 2 3 =>
    rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]
  slice_rhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_rhs 2 3 => 

Proof state:
case a.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
| (𝟙 (M.X ⊗ N.X) ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
    (α_ M.X N.X (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)))).hom
case a.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
| 𝟙 M.X ⊗
    coequalizer.π (N.actRight ⊗ 𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))))
      ((α_ N.X X.X (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X Y.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)))).hom ≫
        (𝟙 N.X ⊗ TensorBimod.actLeft P Q))
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
    (M.actRight ⊗
      𝟙
        (coequalizer (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
          ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft))))
    ((α_ M.X W.X
          (coequalizer (N.actRight ⊗ 𝟙 (tensorBimod P Q).X)
            ((α_ N.X X.X (tensorBimod P Q).X).hom ≫ (𝟙 N.X ⊗ (tensorBimod P Q).actLeft)))).hom ≫
      (𝟙 M.X ⊗ TensorBimod.actLeft N (tensorBimod 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
| (α_ (M.X ⊗ N.X) P.X Q.X).hom

Next tactic:
rw [← MonoidalCategory.tensor_id, associator_naturality]