-- 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
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
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.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))
    (Iso.inv (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.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)
          ((Iso.hom (α_ P.X S.X Q.X) ▷ L.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) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.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) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.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))
    (Iso.hom (α_ P.X S.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X)) ≫
        CategoryStruct.comp (Y :=
          coequalizer (X :=
            tensorObj.{v₁, u₁} (𝒞 := inst) P.X
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X))
            (P.X ◁ Q.actRight ▷ L.X) (P.X ◁ (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
          (CategoryStruct.comp (Y :=
            tensorObj.{v₁, u₁} (𝒞 := inst) P.X
              (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
            (P.X ◁ Iso.inv (α_ S.X Q.X L.X) ≫
              P.X ◁ Q.actLeft ▷ L.X ≫
                P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))
            (Iso.inv
              (PreservesCoequalizer.iso (tensorLeft P.X) (Q.actRight ▷ L.X)
                (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))))
          (coequalizer.desc
            (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) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.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))
have slice_rhs : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.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))
  (Iso.inv (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.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)
        ((Iso.hom (α_ P.X S.X Q.X) ▷ L.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) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.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) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.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))
  (Iso.hom (α_ P.X S.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X)) ≫
      P.X ◁ Iso.inv (α_ S.X Q.X L.X) ≫
        P.X ◁ Q.actLeft ▷ L.X ≫
          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) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (tensorObj.{v₁, u₁} (𝒞 := inst) Q.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: slice_rhs 4 6 => rw [id_tensor_π_preserves_coequalizer_inv_desc]