Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: case a.a
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)
V W X Y Z : Mon C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
| coequalizer.π (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷
      coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
    (PreservesCoequalizer.iso (tensorRight (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)))
          (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
      coequalizer.desc
        ((α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
          M.X ◁
              coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
                ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
                  N.X ◁ TensorBimod.actLeft P Q) ≫
            coequalizer.π
              (M.actRight ▷
                coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                  ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
              ((α_ M.X W.X
                    (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                      ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
                M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)))
        ⋯

pp: case a
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)
V W X Y Z : Mon C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
| (α_ (M.X ⊗ N.X) P.X Q.X).hom

pp: case a.a
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)
V W X Y Z : Mon C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
| (M.X ⊗ N.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)

Tactic:
rw [π_tensor_id_preserves_coequalizer_inv_desc]

Goal state after:
case a.a
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)
V W X Y Z : Mon C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
| (α_ M.X N.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
    M.X ◁
        coequalizer.π (N.actRight ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))
          ((α_ N.X X.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
            N.X ◁ TensorBimod.actLeft P Q) ≫
      coequalizer.π
        (M.actRight ▷
          coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
            ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))
        ((α_ M.X W.X
              (coequalizer (N.actRight ▷ (P.tensorBimod Q).X)
                ((α_ N.X X.X (P.tensorBimod Q).X).hom ≫ N.X ◁ (P.tensorBimod Q).actLeft))).hom ≫
          M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))

case a
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)
V W X Y Z : Mon C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
| (α_ (M.X ⊗ N.X) P.X Q.X).hom

case a.a
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)
V W X Y Z : Mon C
M : Bimod V W
N : Bimod W X
P : Bimod X Y
Q : Bimod Y Z
| (M.X ⊗ N.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft)