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

Tactic:
slice_rhs 2 3 =>
    rw [← whiskerLeft_comp, whiskerLeft_π_actLeft,
      whiskerLeft_comp, whiskerLeft_comp]

Premises used: Bimod.TensorBimod.whiskerLeft_π_actLeft, CategoryTheory.MonoidalCategory.whiskerLeft_comp