theorem comp_whisker_left_bimod {W X Y Z : Mon_ C} (M : Bimod W X) (N : Bimod X Y)
    {P P' : Bimod Y Z} (f : P ⟶ P') :
    tensorHom (𝟙 (M.tensorBimod N)) f =
      (associatorBimod M N P).hom ≫
        tensorHom (𝟙 M) (tensorHom (𝟙 N) f) ≫ (associatorBimod M N P').inv := by
  dsimp [tensorHom, tensorBimod, associatorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  dsimp [TensorBimod.X, AssociatorBimod.hom]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [AssociatorBimod.homAux, AssociatorBimod.inv]
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  rw [tensorRight_map]
  slice_rhs 1 3 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_rhs 3 4 => rw [ι_colimMap, parallelPairHom_app_one]
  slice_rhs 2 3 => rw [← id_tensor_comp, ι_colimMap, parallelPairHom_app_one]
  slice_rhs 3 4 => rw [coequalizer.π_desc]
  dsimp [AssociatorBimod.invAux]
  slice_rhs 2 2 => rw [id_tensor_comp]
  slice_rhs 3 5 => 

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)
W X Y Z : Mon_ C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
| (𝟙 M.X ⊗
      colimit.ι (parallelPair (N.actRight ⊗ 𝟙 P'.X) ((α_ N.X Y.X P'.X).hom ≫ (𝟙 N.X ⊗ P'.actLeft)))
        WalkingParallelPair.one) ≫
    (PreservesCoequalizer.iso (tensorLeft M.X) (N.actRight ⊗ 𝟙 P'.X)
          ((α_ N.X Y.X P'.X).hom ≫ (𝟙 N.X ⊗ P'.actLeft))).inv ≫
      coequalizer.desc
        ((α_ M.X N.X P'.X).inv ≫
          (coequalizer.π (M.actRight ⊗ 𝟙 N.X) ((α_ M.X X.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft)) ⊗ 𝟙 P'.X) ≫
            coequalizer.π (TensorBimod.actRight M N ⊗ 𝟙 P'.X)
              ((α_ (TensorBimod.X M N) Y.X P'.X).hom ≫ (𝟙 (TensorBimod.X M N) ⊗ P'.actLeft)))
        (_ :
          (tensorLeft M.X).map (N.actRight ⊗ 𝟙 P'.X) ≫
              (α_ M.X N.X P'.X).inv ≫
                (coequalizer.π (M.actRight ⊗ 𝟙 N.X) ((α_ M.X X.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft)) ⊗ 𝟙 P'.X) ≫
                  coequalizer.π ((tensorBimod M N).actRight ⊗ 𝟙 P'.X)
                    ((α_ (tensorBimod M N).X Y.X P'.X).hom ≫ (𝟙 (tensorBimod M N).X ⊗ P'.actLeft)) =
            (tensorLeft M.X).map ((α_ N.X Y.X P'.X).hom ≫ (𝟙 N.X ⊗ P'.actLeft)) ≫
              (α_ M.X N.X P'.X).inv ≫
                (coequalizer.π (M.actRight ⊗ 𝟙 N.X) ((α_ M.X X.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft)) ⊗ 𝟙 P'.X) ≫
                  coequalizer.π ((tensorBimod M N).actRight ⊗ 𝟙 P'.X)
                    ((α_ (tensorBimod M N).X Y.X P'.X).hom ≫ (𝟙 (tensorBimod M N).X ⊗ P'.actLeft)))
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)
W X Y Z : Mon_ C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
| (α_ M.X N.X P.X).hom
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)
W X Y Z : Mon_ C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
| 𝟙 M.X ⊗ 𝟙 N.X ⊗ f.hom

Next tactic:
rw [id_tensor_π_preserves_coequalizer_inv_desc]