-- 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) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) L.X ⟶
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
          (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
        (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.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
              (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
            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)))
        (Iso.hom (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
            P.X ◁ TensorBimod.actLeft Q L :
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
              (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
            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))))
    (Iso.hom (α_ P.X Q.X T.X) ▷ L.X ≫
        Iso.hom (α_ P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ≫
          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.hom (α_ Q.X T.X L.X) ≫ P.X ◁ Q.X ◁ L.actLeft) ≫
                P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ 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) ⟶
                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)))
            (coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))
                (Iso.hom (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
                  P.X ◁ TensorBimod.actLeft Q L) :
              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)) ⟶
                coequalizer (X :=
                  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                    (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
                  (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.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
                    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                        (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
                      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)))
                  (Iso.hom (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
                      P.X ◁ TensorBimod.actLeft Q L :
                    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                        (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
                      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)))) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) L.X ⟶
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
            (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
          (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.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
              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)))
          (Iso.hom (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
              P.X ◁ TensorBimod.actLeft Q L :
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
              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))))
    (coequalizer.π (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ T.X ▷ 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 :=
            coequalizer (X :=
              tensorObj.{v₁, u₁} (𝒞 := inst)
                (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) Q.X) L.X)
              (P.actRight ▷ Q.X ▷ L.X) ((Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X))
            (Iso.inv
              (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X)
                (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
            (coequalizer.desc
              (Iso.hom (α_ P.X Q.X L.X) ≫
                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 ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
                    tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X) ⟶
                      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)))
                  (coequalizer.π
                      (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))
                      (Iso.hom
                          (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
                        P.X ◁ TensorBimod.actLeft Q L) :
                    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)) ⟶
                      coequalizer (X :=
                        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                          (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
                        (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.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                              (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
                            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)))
                        (Iso.hom
                              (α_ P.X S.X
                                (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
                            P.X ◁ TensorBimod.actLeft Q L :
                          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                              (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
                            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)))))
              ⋯) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) L.X ⟶
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
            (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
          (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.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
              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)))
          (Iso.hom (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
              P.X ◁ TensorBimod.actLeft Q L :
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
              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))))
have slice_rhs : Eq (α :=
  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) L.X ⟶
    coequalizer (X :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
        (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
      (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.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
            (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
          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)))
      (Iso.hom (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
          P.X ◁ TensorBimod.actLeft Q L :
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
            (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
          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))))
  (Iso.hom (α_ P.X Q.X T.X) ▷ L.X ≫
      Iso.hom (α_ P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X T.X) L.X) ≫
        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.hom (α_ Q.X T.X L.X) ≫ P.X ◁ Q.X ◁ L.actLeft) ≫
              P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ 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) ⟶
              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)))
          (coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))
              (Iso.hom (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
                P.X ◁ TensorBimod.actLeft Q L) :
            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)) ⟶
              coequalizer (X :=
                tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                  (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
                (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.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
                  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                      (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
                    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)))
                (Iso.hom (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
                    P.X ◁ TensorBimod.actLeft Q L :
                  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                      (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
                    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)))) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) L.X ⟶
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
          (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
        (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.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
              (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
            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)))
        (Iso.hom (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
            P.X ◁ TensorBimod.actLeft Q L :
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
              (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
            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))))
  (CategoryStruct.comp (Y :=
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) Q.X)
          L.X)
        (P.actRight ▷ Q.X ▷ L.X) ((Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft) ▷ L.X))
      (((Iso.hom (α_ (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X L.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) 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
          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) (Iso.hom (α_ P.X S.X Q.X) ≫ P.X ◁ Q.actLeft)))
      (coequalizer.desc
        (Iso.hom (α_ P.X Q.X L.X) ≫
          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 ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
              tensorObj.{v₁, u₁} (𝒞 := inst) P.X (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X) ⟶
                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)))
            (coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))
                (Iso.hom (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
                  P.X ◁ TensorBimod.actLeft Q L) :
              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)) ⟶
                coequalizer (X :=
                  tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                    (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
                  (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.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
                    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                        (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
                      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)))
                  (Iso.hom (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
                      P.X ◁ TensorBimod.actLeft Q L :
                    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
                        (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
                      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)))))
        sorry) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X Q.X) T.X) L.X ⟶
      coequalizer (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
          (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
        (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.actRight ▷ coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) :
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
              (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
            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)))
        (Iso.hom (α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft))) ≫
            P.X ◁ TensorBimod.actLeft Q L :
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X)
              (coequalizer (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) ⟶
            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))))
Tactic: slice_rhs 1 2 => rw [associator_naturality_left]