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 ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ▷ Q.X ≫
    ((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
        coequalizer.desc
          ((α_ M.X N.X P.X).hom ≫
            M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫
              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

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
| (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

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.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
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))
              ⋯)
          ⋯)
      ⋯ ⋯)

Tactic:
rw [← comp_whiskerRight, π_tensor_id_preserves_coequalizer_inv_desc, comp_whiskerRight,
      comp_whiskerRight]

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

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
| (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

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.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))
    ⋯

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
| 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))
              ⋯)
          ⋯)
      ⋯ ⋯)