/-- Tensor product of two morphisms of bimodule objects. -/
@[simps]
noncomputable def tensorHom {X Y Z : Mon_ C} {M₁ M₂ : Bimod X Y} {N₁ N₂ : Bimod Y Z} (f : M₁ ⟶ M₂)
    (g : N₁ ⟶ N₂) : M₁.tensorBimod N₁ ⟶ M₂.tensorBimod N₂ where
  hom :=
    colimMap
      (parallelPairHom _ _ _ _ ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
        (by
          rw [← tensor_comp, ← tensor_comp, Hom.right_act_hom, Category.id_comp, Category.comp_id])
        (by
          slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
          slice_rhs 1 2 => rw [associator_naturality]
          slice_rhs 2 3 => rw [← tensor_comp, Category.comp_id]))
  left_act_hom := by
    refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
    dsimp
    slice_lhs 1 2 => rw [TensorBimod.id_tensor_π_actLeft]
    slice_lhs 3 4 => rw [ι_colimMap, parallelPairHom_app_one]
    slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
    slice_rhs 1 2 => rw [← id_tensor_comp, ι_colimMap, parallelPairHom_app_one, id_tensor_comp]
    slice_rhs 2 3 => rw [TensorBimod.id_tensor_π_actLeft]
    slice_rhs 1 2 => rw [associator_inv_naturality]
    slice_rhs 2 3 => rw [← tensor_comp, Category.comp_id]
  right_act_hom := by
    refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
    dsimp
    slice_lhs 1 2 => rw [TensorBimod.π_tensor_id_actRight]
    

Proof state:
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)
X Y Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
⊢ ((α_ M₁.X N₁.X Z.X).hom ≫
        (𝟙 M₁.X ⊗ N₁.actRight) ≫
          coequalizer.π (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))) ≫
      colimMap
        (parallelPairHom (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) (M₂.actRight ⊗ 𝟙 N₂.X)
          ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)) ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
          (_ : (M₁.actRight ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (M₂.actRight ⊗ 𝟙 N₂.X))
          (_ :
            ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫ (f.hom ⊗ g.hom) =
              ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft))) =
    (coequalizer.π (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ⊗ 𝟙 Z.X) ≫
      (colimMap
            (parallelPairHom (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))
              (M₂.actRight ⊗ 𝟙 N₂.X) ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)) ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom)
              (f.hom ⊗ g.hom)
              (_ : (M₁.actRight ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (M₂.actRight ⊗ 𝟙 N₂.X))
              (_ :
                ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫ (f.hom ⊗ g.hom) =
                  ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft))) ⊗
          𝟙 Z.X) ≫
        TensorBimod.actRight M₂ N₂

Next tactic:
slice_lhs 3 4 => rw [ι_colimMap, parallelPairHom_app_one]