theorem hom_left_act_hom' :
    ((P.tensorBimod Q).tensorBimod L).actLeft ≫ hom P Q L =
      (𝟙 R.X ⊗ hom P Q L) ≫ (P.tensorBimod (Q.tensorBimod L)).actLeft := by
  dsimp; dsimp [hom, homAux]
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  rw [tensorLeft_map]
  slice_lhs 1 2 => rw [TensorBimod.id_tensor_π_actLeft]
  slice_lhs 3 4 => rw [coequalizer.π_desc]
  slice_rhs 1 2 => rw [← id_tensor_comp, coequalizer.π_desc, id_tensor_comp]
  refine' (cancel_epi ((tensorRight _ ⋙ tensorLeft _).map (coequalizer.π _ _))).1 _
  dsimp; dsimp [TensorBimod.X]
  slice_lhs 1 2 => rw [associator_inv_naturality]
  slice_lhs 2 3 =>
    rw [← comp_tensor_id, TensorBimod.id_tensor_π_actLeft, comp_tensor_id, comp_tensor_id]
  slice_lhs 4 6 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_lhs 3 4 => rw [associator_naturality]
  slice_lhs 4 5 => rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor]
  slice_rhs 1 3 =>
    

Proof state:
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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| (𝟙 R.X ⊗ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 L.X) ≫
    (𝟙 R.X ⊗
        (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ⊗ 𝟙 Q.X)
            ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))).inv) ≫
      (𝟙 R.X ⊗
        coequalizer.desc
          ((α_ P.X Q.X L.X).hom ≫
            (𝟙 P.X ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft))) ≫
              coequalizer.π
                (P.actRight ⊗ 𝟙 (coequalizer (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft))))
                ((α_ P.X S.X (coequalizer (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft)))).hom ≫
                  (𝟙 P.X ⊗ TensorBimod.actLeft Q L)))
          (_ :
            (tensorRight L.X).map (P.actRight ⊗ 𝟙 Q.X) ≫
                (α_ P.X Q.X L.X).hom ≫
                  (𝟙 P.X ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft))) ≫
                    coequalizer.π (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
                      ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q L).actLeft)) =
              (tensorRight L.X).map ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ≫
                (α_ P.X Q.X L.X).hom ≫
                  (𝟙 P.X ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft))) ≫
                    coequalizer.π (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
                      ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q L).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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| TensorBimod.actLeft P (tensorBimod Q L)

Next tactic:
rw [← id_tensor_comp, ← id_tensor_comp, π_tensor_id_preserves_coequalizer_inv_desc,
      id_tensor_comp, id_tensor_comp]