Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: case a.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
| (M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) ▷ Q.X ≫
    (α_ M.X (coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) Q.X).hom

pp: case a.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
| M.X ◁
    coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
      ((α_ (coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) Y.X Q.X).hom ≫
        coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ◁ Q.actLeft)

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 ▷
      coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
        ((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))
    ((α_ M.X W.X
          (coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
            ((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))).hom ≫
      M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)

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
| colimMap
    (parallelPairHom
      (M.actRight ▷
        coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
          ((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))
      ((α_ M.X W.X
            (coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
              ((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))).hom ≫
        M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
      (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))
      ((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 ▷ 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))
              ⋯)
          ⋯)
      (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 ▷ 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))
              ⋯)
          ⋯)
      ⋯ ⋯)

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

Tactic:
rw [associator_naturality_middle]

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

case a.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
| M.X ◁
    coequalizer.π (TensorBimod.actRight N P ▷ Q.X)
      ((α_ (coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)) Y.X Q.X).hom ≫
        coequalizer (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ◁ Q.actLeft)

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 ▷
      coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
        ((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))
    ((α_ M.X W.X
          (coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
            ((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))).hom ≫
      M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)

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
| colimMap
    (parallelPairHom
      (M.actRight ▷
        coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
          ((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))
      ((α_ M.X W.X
            (coequalizer ((N.tensorBimod P).actRight ▷ Q.X)
              ((α_ (N.tensorBimod P).X Y.X Q.X).hom ≫ (N.tensorBimod P).X ◁ Q.actLeft))).hom ≫
        M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)
      (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))
      ((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 ▷ 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))
              ⋯)
          ⋯)
      (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 ▷ 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))
              ⋯)
          ⋯)
      ⋯ ⋯)

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