-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Bimod (goal)
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)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.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) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
          T.X)
        (Iso.inv (α_ R.X P.X Q.X) ▷ T.X ≫
            P.actLeft ▷ Q.X ▷ T.X ≫
              coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ T.X :
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X))
              T.X ⟶
            tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
              T.X)
        (actRight P Q : tensorObj.{v₁, u₁} (𝒞 := inst) (X P Q) T.X ⟶ X P Q) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.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)
          (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)))
          T.X)
        ((R.X ◁ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) ▷ T.X :
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X))
              T.X ⟶
            tensorObj.{v₁, u₁} (𝒞 := inst)
              (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)))
              T.X)
        (Iso.hom (α_ R.X (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.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 ◁ actRight P Q :
                tensorObj.{v₁, u₁} (𝒞 := inst) R.X
                    (tensorObj.{v₁, u₁} (𝒞 := inst)
                      (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.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
                (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
              T.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 (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.X ⟶
        coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
have slice_lhs : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.X ⟶
    coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
  (Iso.inv (α_ R.X P.X Q.X) ▷ T.X ≫
      P.actLeft ▷ Q.X ▷ T.X ≫
        Iso.hom (α_ P.X Q.X T.X) ≫
          P.X ◁ Q.actRight ≫ 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 (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.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)
        (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)))
        T.X)
      ((R.X ◁ coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) ▷ T.X :
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) R.X (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X))
            T.X ⟶
          tensorObj.{v₁, u₁} (𝒞 := inst)
            (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)))
            T.X)
      (Iso.hom (α_ R.X (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.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 ◁ actRight P Q :
              tensorObj.{v₁, u₁} (𝒞 := inst) R.X
                  (tensorObj.{v₁, u₁} (𝒞 := inst)
                    (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)) T.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
              (coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
            T.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 (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X)) T.X ⟶
      coequalizer (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft))
Tactic: slice_lhs 3 4 => rw [π_tensor_id_actRight]