Module: Mathlib.CategoryTheory.Monoidal.Bimod

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

Tactic:
rw [π_tensor_id_preserves_coequalizer_inv_desc]

Goal state after:
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_ (TensorBimod.X M N) P.X Q.X).hom ≫
    TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
      coequalizer.π (TensorBimod.actRight M N ▷ TensorBimod.X P Q)
        ((α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)).hom ≫ TensorBimod.X M N ◁ 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
| coequalizer.desc
    ((PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X)
          ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫
      coequalizer.desc
        ((α_ M.X N.X (TensorBimod.X P Q)).hom ≫
          M.X ◁
              coequalizer.π (N.actRight ▷ TensorBimod.X P Q)
                ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫
            coequalizer.π (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)))
        ⋯)
    ⋯