-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
A B : Mon.{v₁, u₁} C
M : Bimod A B
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon.{v₁, u₁} C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
rw :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
      coequalizer (TensorBimod.actRight P Q ▷ L.X)
        (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
          coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft))
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
          L.X)
        ((Iso.inv (α_ P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ≫
              Iso.inv (α_ P.X Q.X T.X) ▷ L.X ≫ (Iso.hom (α_ P.X Q.X T.X) ≫ P.X ◁ Q.actRight) ▷ L.X) ≫
            coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X :
          tensorObj.{v₁, u₁} (𝒞 := inst) P.X
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
            tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
              L.X)
        (coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
            (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
              coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) :
          tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
              L.X ⟶
            coequalizer (TensorBimod.actRight P Q ▷ L.X)
              (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
                coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft)) :
      tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
        coequalizer (TensorBimod.actRight P Q ▷ L.X)
          (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
            coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft))
    (P.X ◁ (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ≫
        Iso.inv (α_ P.X Q.X L.X) ≫
          CategoryStruct.comp (Y :=
            tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
              L.X)
            (coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X :
              tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) L.X ⟶
                tensorObj.{v₁, u₁} (𝒞 := inst)
                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) L.X)
            (coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
                (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
                  coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) :
              tensorObj.{v₁, u₁} (𝒞 := inst)
                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) L.X ⟶
                coequalizer (TensorBimod.actRight P Q ▷ L.X)
                  (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
                    coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft)) :
      tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
        coequalizer (TensorBimod.actRight P Q ▷ L.X)
          (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
            coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
      coequalizer (TensorBimod.actRight P Q ▷ L.X)
        (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
          coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft))
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
          L.X)
        ((Iso.inv (α_ P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ≫ (P.X ◁ Q.actRight) ▷ L.X) ≫
            coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X :
          tensorObj.{v₁, u₁} (𝒞 := inst) P.X
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
            tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
              L.X)
        (coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
            (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
              coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) :
          tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
              L.X ⟶
            coequalizer (TensorBimod.actRight P Q ▷ L.X)
              (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
                coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft)) :
      tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
        coequalizer (TensorBimod.actRight P Q ▷ L.X)
          (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
            coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft))
    (P.X ◁ (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ≫
        Iso.inv (α_ P.X Q.X L.X) ≫
          CategoryStruct.comp (Y :=
            tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
              L.X)
            (coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X :
              tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) L.X ⟶
                tensorObj.{v₁, u₁} (𝒞 := inst)
                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) L.X)
            (coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
                (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
                  coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) :
              tensorObj.{v₁, u₁} (𝒞 := inst)
                  (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) L.X ⟶
                coequalizer (TensorBimod.actRight P Q ▷ L.X)
                  (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
                    coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft)) :
      tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ⟶
        coequalizer (TensorBimod.actRight P Q ▷ L.X)
          (Iso.hom (α_ (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.X L.X) ≫
            coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ◁ L.actLeft))

Tactic:
rw [← Iso.inv_hom_id_assoc (α_ _ _ _) (P.X ◁ Q.actRight), comp_whiskerRight]

Premises used: CategoryTheory.MonoidalCategoryStruct.whiskerLeft, CategoryTheory.MonoidalCategory.comp_whiskerRight, CategoryTheory.Iso.inv_hom_id_assoc, Bimod.X, Bimod.actRight, CategoryTheory.MonoidalCategoryStruct.associator