-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
theorem inv_hom_id : inv P Q L ≫ hom P Q L = 𝟙 _

Proof state:
case h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
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
⊢ ((PreservesCoequalizer.iso (tensorLeft P.X) (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)).inv ≫
        coequalizer.desc
          ((α_ P.X Q.X L.X).inv ≫
            coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
              coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
                ((α_ (TensorBimod.X P Q) T.X L.X).hom ≫ TensorBimod.X P Q ◁ L.actLeft))
          ⋯) ≫
      coequalizer.desc
        ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
          coequalizer.desc
            ((α_ P.X Q.X L.X).hom ≫
              P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
                coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
                  ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
            ⋯)
        ⋯ =
    coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X)
        ((α_ P.X S.X (Q.tensorBimod L).X).hom ≫ P.X ◁ (Q.tensorBimod L).actLeft) ≫
      𝟙 (TensorBimod.X P (Q.tensorBimod L))

Tactic:
refine (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 ?_

Result:
case h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
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
⊢ (tensorLeft P.X).map (coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫
      ((PreservesCoequalizer.iso (tensorLeft P.X) (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)).inv ≫
          coequalizer.desc
            ((α_ P.X Q.X L.X).inv ≫
              coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
                coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
                  ((α_ (TensorBimod.X P Q) T.X L.X).hom ≫ TensorBimod.X P Q ◁ L.actLeft))
            ⋯) ≫
        coequalizer.desc
          ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
                ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
            coequalizer.desc
              ((α_ P.X Q.X L.X).hom ≫
                P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
                  coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
                    ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
              ⋯)
          ⋯ =
    (tensorLeft P.X).map (coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫
      coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X)
          ((α_ P.X S.X (Q.tensorBimod L).X).hom ≫ P.X ◁ (Q.tensorBimod L).actLeft) ≫
        𝟙 (TensorBimod.X P (Q.tensorBimod L))

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
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
⊢ coequalizer.desc
        ((PreservesCoequalizer.iso (tensorLeft P.X) (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)).inv ≫
          coequalizer.desc
            ((α_ P.X Q.X L.X).inv ≫
              coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
                coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
                  ((α_ (TensorBimod.X P Q) T.X L.X).hom ≫ TensorBimod.X P Q ◁ L.actLeft))
            ⋯)
        ⋯ ≫
      coequalizer.desc
        ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
          coequalizer.desc
            ((α_ P.X Q.X L.X).hom ≫
              P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
                coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
                  ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
            ⋯)
        ⋯ =
    𝟙 (TensorBimod.X P (Q.tensorBimod L))

Tactic:
apply coequalizer.hom_ext

Result:
case h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
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
⊢ coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X)
        ((α_ P.X S.X (Q.tensorBimod L).X).hom ≫ P.X ◁ (Q.tensorBimod L).actLeft) ≫
      coequalizer.desc
          ((PreservesCoequalizer.iso (tensorLeft P.X) (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)).inv ≫
            coequalizer.desc
              ((α_ P.X Q.X L.X).inv ≫
                coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
                  coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
                    ((α_ (TensorBimod.X P Q) T.X L.X).hom ≫ TensorBimod.X P Q ◁ L.actLeft))
              ⋯)
          ⋯ ≫
        coequalizer.desc
          ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
                ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
            coequalizer.desc
              ((α_ P.X Q.X L.X).hom ≫
                P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
                  coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
                    ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
              ⋯)
          ⋯ =
    coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X)
        ((α_ P.X S.X (Q.tensorBimod L).X).hom ≫ P.X ◁ (Q.tensorBimod L).actLeft) ≫
      𝟙 (TensorBimod.X P (Q.tensorBimod L))

Proof state:
case h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
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
⊢ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
      coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))
        ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫
          P.X ◁ TensorBimod.actLeft Q L) =
    P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
      coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X)
        ((α_ P.X S.X (Q.tensorBimod L).X).hom ≫ P.X ◁ (Q.tensorBimod L).actLeft)

Tactic:
rfl

Result:
no goals