Theorem: Bimod.TensorBimod.left_assoc'
File: Mathlib/CategoryTheory/Monoidal/Bimod.lean

Proof steps:
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ R.mul ▷ X P Q ≫ actLeft P Q = (α_ R.X R.X (X P Q)).hom ≫ R.X ◁ actLeft P Q ≫ actLeft P Q
state_after: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (tensorLeft (R.X ⊗ R.X)).map (coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫
      R.mul ▷ X P Q ≫ actLeft P Q =
    (tensorLeft (R.X ⊗ R.X)).map (coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫
      (α_ R.X R.X (X P Q)).hom ≫ R.X ◁ actLeft P Q ≫ actLeft P Q
tactic: refine (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 ?_
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (tensorLeft (R.X ⊗ R.X)).map (coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫
      R.mul ▷ X P Q ≫ actLeft P Q =
    (tensorLeft (R.X ⊗ R.X)).map (coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫
      (α_ R.X R.X (X P Q)).hom ≫ R.X ◁ actLeft P Q ≫ actLeft P Q
state_after: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (R.X ⊗ R.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
      R.mul ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ actLeft P Q =
    (R.X ⊗ R.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
      (α_ R.X R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
        R.X ◁ actLeft P Q ≫ actLeft P Q
tactic: dsimp [X]
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (R.X ⊗ R.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
      R.mul ▷ coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ actLeft P Q =
    (R.X ⊗ R.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
      (α_ R.X R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
        R.X ◁ actLeft P Q ≫ actLeft P Q
state_after: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (R.mul ▷ (P.X ⊗ Q.X) ≫ R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫
      actLeft P Q =
    (R.X ⊗ R.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
      (α_ R.X R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
        R.X ◁ actLeft P Q ≫ actLeft P Q
tactic: slice_lhs 1 2 => rw [whisker_exchange]
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (R.mul ▷ (P.X ⊗ Q.X) ≫ R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫
      actLeft P Q =
    (R.X ⊗ R.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
      (α_ R.X R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
        R.X ◁ actLeft P Q ≫ actLeft P Q
state_after: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ R.mul ▷ (P.X ⊗ Q.X) ≫
      (α_ R.X P.X Q.X).inv ≫
        P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (R.X ⊗ R.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
      (α_ R.X R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
        R.X ◁ actLeft P Q ≫ actLeft P Q
tactic: slice_lhs 2 3 => rw [whiskerLeft_π_actLeft]
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ R.mul ▷ (P.X ⊗ Q.X) ≫
      (α_ R.X P.X Q.X).inv ≫
        P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (R.X ⊗ R.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
      (α_ R.X R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
        R.X ◁ actLeft P Q ≫ actLeft P Q
state_after: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (((α_ (R.X ⊗ R.X) P.X Q.X).inv ≫ R.mul ▷ P.X ▷ Q.X) ≫ P.actLeft ▷ Q.X) ≫
      coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (R.X ⊗ R.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
      (α_ R.X R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
        R.X ◁ actLeft P Q ≫ actLeft P Q
tactic: slice_lhs 1 2 => rw [associator_inv_naturality_left]
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (((α_ (R.X ⊗ R.X) P.X Q.X).inv ≫ R.mul ▷ P.X ▷ Q.X) ≫ P.actLeft ▷ Q.X) ≫
      coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (R.X ⊗ R.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
      (α_ R.X R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
        R.X ◁ actLeft P Q ≫ actLeft P Q
state_after: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
      ((α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft) ▷ Q.X ≫ P.actLeft ▷ Q.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (R.X ⊗ R.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
      (α_ R.X R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
        R.X ◁ actLeft P Q ≫ actLeft P Q
tactic: slice_lhs 2 3 => rw [← comp_whiskerRight, left_assoc, comp_whiskerRight, comp_whiskerRight]
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
      ((α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft) ▷ Q.X ≫ P.actLeft ▷ Q.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (R.X ⊗ R.X) ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫
      (α_ R.X R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft))).hom ≫
        R.X ◁ actLeft P Q ≫ actLeft P Q
state_after: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
      ((α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft) ▷ Q.X ≫ P.actLeft ▷ Q.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (((α_ R.X R.X (P.X ⊗ Q.X)).hom ≫
          R.X ◁ R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫
        R.X ◁ actLeft P Q) ≫
      actLeft P Q
tactic: slice_rhs 1 2 => rw [associator_naturality_right]
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
      ((α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft) ▷ Q.X ≫ P.actLeft ▷ Q.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (((α_ R.X R.X (P.X ⊗ Q.X)).hom ≫
          R.X ◁ R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫
        R.X ◁ actLeft P Q) ≫
      actLeft P Q
state_after: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
      ((α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft) ▷ Q.X ≫ P.actLeft ▷ Q.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (α_ R.X R.X (P.X ⊗ Q.X)).hom ≫
      (R.X ◁ (α_ R.X P.X Q.X).inv ≫
          R.X ◁ P.actLeft ▷ Q.X ≫ R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫
        actLeft P Q
tactic: slice_rhs 2 3 =>
    rw [← MonoidalCategory.whiskerLeft_comp, whiskerLeft_π_actLeft,
      MonoidalCategory.whiskerLeft_comp, MonoidalCategory.whiskerLeft_comp]
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
      ((α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft) ▷ Q.X ≫ P.actLeft ▷ Q.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (α_ R.X R.X (P.X ⊗ Q.X)).hom ≫
      (R.X ◁ (α_ R.X P.X Q.X).inv ≫
          R.X ◁ P.actLeft ▷ Q.X ≫ R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫
        actLeft P Q
state_after: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
      ((α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft) ▷ Q.X ≫ P.actLeft ▷ Q.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (α_ R.X R.X (P.X ⊗ Q.X)).hom ≫
      R.X ◁ (α_ R.X P.X Q.X).inv ≫
        R.X ◁ P.actLeft ▷ Q.X ≫
          (α_ R.X P.X Q.X).inv ≫
            P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)
tactic: slice_rhs 4 5 => rw [whiskerLeft_π_actLeft]
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
      ((α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft) ▷ Q.X ≫ P.actLeft ▷ Q.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (α_ R.X R.X (P.X ⊗ Q.X)).hom ≫
      R.X ◁ (α_ R.X P.X Q.X).inv ≫
        R.X ◁ P.actLeft ▷ Q.X ≫
          (α_ R.X P.X Q.X).inv ≫
            P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)
state_after: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
      ((α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft) ▷ Q.X ≫ P.actLeft ▷ Q.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (α_ R.X R.X (P.X ⊗ Q.X)).hom ≫
      R.X ◁ (α_ R.X P.X Q.X).inv ≫
        (((α_ R.X (R.X ⊗ P.X) Q.X).inv ≫ (R.X ◁ P.actLeft) ▷ Q.X) ≫ P.actLeft ▷ Q.X) ≫
          coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)
tactic: slice_rhs 3 4 => rw [associator_inv_naturality_middle]
state_before: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon_ C
P : Bimod R S
Q : Bimod S T
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
⊢ (α_ (R.X ⊗ R.X) P.X Q.X).inv ≫
      ((α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft) ▷ Q.X ≫ P.actLeft ▷ Q.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =
    (α_ R.X R.X (P.X ⊗ Q.X)).hom ≫
      R.X ◁ (α_ R.X P.X Q.X).inv ≫
        (((α_ R.X (R.X ⊗ P.X) Q.X).inv ≫ (R.X ◁ P.actLeft) ▷ Q.X) ≫ P.actLeft ▷ Q.X) ≫
          coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)
state_after: no goals
tactic: coherence