/-- The underlying morphism of the inverse component of the associator isomorphism. -/
noncomputable def inv :
    (P.tensorBimod (Q.tensorBimod L)).X ⟶ ((P.tensorBimod Q).tensorBimod L).X :=
  coequalizer.desc (invAux P Q L)
    (by
      dsimp [invAux]
      refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
      dsimp [TensorBimod.X]
      slice_lhs 1 2 => rw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
      slice_lhs 2 4 => rw [id_tensor_π_preserves_coequalizer_inv_desc]
      slice_lhs 1 2 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
      slice_lhs 2 3 => rw [← comp_tensor_id, coequalizer.condition, comp_tensor_id, comp_tensor_id]
      

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

Next tactic:
slice_rhs 1 2 => rw [← MonoidalCategory.tensor_id, associator_naturality]