Module: Mathlib.CategoryTheory.Monoidal.Bimod

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

pp: case a.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
| colimMap
    (parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
      ((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
      (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
      ((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
      ((M.X ⊗ W.X) ◁
        coequalizer.desc
          ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
                ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
            coequalizer.desc
              ((α_ N.X P.X Q.X).hom ≫
                N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
                  coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
                    ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
              ⋯)
          ⋯)
      (M.X ◁
        coequalizer.desc
          ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
                ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
            coequalizer.desc
              ((α_ N.X P.X Q.X).hom ≫
                N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
                  coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
                    ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P 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).hom ▷ Q.X

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 ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) ▷ Q.X

Tactic:
rw [π_tensor_id_preserves_coequalizer_inv_desc]

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

case a.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
| colimMap
    (parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q)
      ((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
      (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q))
      ((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))
      ((M.X ⊗ W.X) ◁
        coequalizer.desc
          ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
                ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
            coequalizer.desc
              ((α_ N.X P.X Q.X).hom ≫
                N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
                  coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
                    ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q))
              ⋯)
          ⋯)
      (M.X ◁
        coequalizer.desc
          ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X)
                ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫
            coequalizer.desc
              ((α_ N.X P.X Q.X).hom ≫
                N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
                  coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
                    ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P 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).hom ▷ Q.X

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 ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) ▷ Q.X