Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: 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)
W X Y Z : Mon C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
⊢ coequalizer.π
        (M.actRight ▷
          { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
              actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X)
        ((α_ M.X X.X
              { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
                  actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯,
                  middle_assoc := ⋯ }.X).hom ≫
          M.X ◁
            { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
                actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯,
                middle_assoc := ⋯ }.actLeft) ≫
      (whiskerRight f
          { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
            actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }).hom =
    coequalizer.π
        (M.actRight ▷
          { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
              actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X)
        ((α_ M.X X.X
              { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
                  actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯,
                  middle_assoc := ⋯ }.X).hom ≫
          M.X ◁
            { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
                actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯,
                middle_assoc := ⋯ }.actLeft) ≫
      ((isoOfIso
              { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯
              ⋯).inv ≫
          whiskerRight (whiskerRight f N) P ≫
            (isoOfIso
                { hom := AssociatorBimod.hom M' N P, inv := AssociatorBimod.inv M' N P, hom_inv_id := ⋯,
                  inv_hom_id := ⋯ }
                ⋯ ⋯).hom).hom

Tactic:
dsimp

Goal state after:
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)
W X Y Z : Mon C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
⊢ coequalizer.π (M.actRight ▷ TensorBimod.X N P)
        ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) ≫
      colimMap
        (parallelPairHom (M.actRight ▷ TensorBimod.X N P)
          ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) (M'.actRight ▷ TensorBimod.X N P)
          ((α_ M'.X X.X (TensorBimod.X N P)).hom ≫ M'.X ◁ TensorBimod.actLeft N P) (f.hom ▷ X.X ▷ TensorBimod.X N P)
          (f.hom ▷ TensorBimod.X N P) ⋯ ⋯) =
    coequalizer.π (M.actRight ▷ TensorBimod.X N P)
        ((α_ M.X X.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P) ≫
      AssociatorBimod.inv M N P ≫
        colimMap
            (parallelPairHom (TensorBimod.actRight M N ▷ P.X)
              ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft) (TensorBimod.actRight M' N ▷ P.X)
              ((α_ (TensorBimod.X M' N) Y.X P.X).hom ≫ TensorBimod.X M' N ◁ P.actLeft)
              (colimMap
                    (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M'.actRight ▷ N.X)
                      ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
                  Y.X ▷
                P.X)
              (colimMap
                  (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M'.actRight ▷ N.X)
                    ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
                P.X)
              ⋯ ⋯) ≫
          AssociatorBimod.hom M' N P