-- 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
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
slice_lhs :
  Eq (α :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight U.X) (tensorObj.{v₁, u₁} (𝒞 := inst) (P.tensorBimod Q).X L.X) ⟶
      TensorBimod.X P (Q.tensorBimod L))
    (Iso.hom (α_ (P.tensorBimod Q).X L.X U.X) ≫
        (P.tensorBimod Q).X ◁ L.actRight ≫
          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 ▷ TensorBimod.X Q L)
                      (Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ 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) (TensorBimod.X Q L))
                        (P.actRight ▷ TensorBimod.X Q L)
                        (Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ P.X ◁ TensorBimod.actLeft Q L)))
              ⋯) :
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight U.X) (tensorObj.{v₁, u₁} (𝒞 := inst) (P.tensorBimod Q).X L.X) ⟶
        TensorBimod.X P (Q.tensorBimod L))
    (CategoryStruct.comp (Y :=
        Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight U.X)
          (coequalizer ((P.tensorBimod Q).actRight ▷ L.X)
            (Iso.hom (α_ (P.tensorBimod Q).X T.X L.X) ≫ (P.tensorBimod Q).X ◁ L.actLeft)))
        (coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
            (Iso.hom (α_ (P.tensorBimod Q).X T.X L.X) ≫ (P.tensorBimod Q).X ◁ L.actLeft) ▷
          U.X)
        (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X P (Q.tensorBimod L)) U.X)
          (coequalizer.desc
                (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 ▷ TensorBimod.X Q L)
                            (Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ 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)
                                (TensorBimod.X Q L))
                              (P.actRight ▷ TensorBimod.X Q L)
                              (Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ P.X ◁ TensorBimod.actLeft Q L)))
                    ⋯))
                ⋯ ▷
              U.X :
            tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X (P.tensorBimod Q) L) U.X ⟶
              tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X P (Q.tensorBimod L)) U.X)
          (TensorBimod.actRight P (Q.tensorBimod L) :
            tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X P (Q.tensorBimod L)) U.X ⟶
              TensorBimod.X P (Q.tensorBimod L))) :
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight U.X) (tensorObj.{v₁, u₁} (𝒞 := inst) (P.tensorBimod Q).X L.X) ⟶
        TensorBimod.X P (Q.tensorBimod L))
⊢ Eq (α :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight U.X) (tensorObj.{v₁, u₁} (𝒞 := inst) (P.tensorBimod Q).X L.X) ⟶
      TensorBimod.X P (Q.tensorBimod L))
    (CategoryStruct.comp (Y := TensorBimod.X (P.tensorBimod Q) L)
        (Iso.hom (α_ (P.tensorBimod Q).X L.X U.X) ≫
          (P.tensorBimod Q).X ◁ L.actRight ≫
            coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
              (Iso.hom (α_ (P.tensorBimod Q).X T.X L.X) ≫ (P.tensorBimod Q).X ◁ L.actLeft))
        (coequalizer.desc
          (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 ▷ TensorBimod.X Q L)
                      (Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ 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) (TensorBimod.X Q L))
                        (P.actRight ▷ TensorBimod.X Q L)
                        (Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ P.X ◁ TensorBimod.actLeft Q L)))
              ⋯))
          ⋯) :
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight U.X) (tensorObj.{v₁, u₁} (𝒞 := inst) (P.tensorBimod Q).X L.X) ⟶
        TensorBimod.X P (Q.tensorBimod L))
    (CategoryStruct.comp (Y :=
        Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight U.X)
          (coequalizer ((P.tensorBimod Q).actRight ▷ L.X)
            (Iso.hom (α_ (P.tensorBimod Q).X T.X L.X) ≫ (P.tensorBimod Q).X ◁ L.actLeft)))
        (coequalizer.π ((P.tensorBimod Q).actRight ▷ L.X)
            (Iso.hom (α_ (P.tensorBimod Q).X T.X L.X) ≫ (P.tensorBimod Q).X ◁ L.actLeft) ▷
          U.X)
        (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X P (Q.tensorBimod L)) U.X)
          (coequalizer.desc
                (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 ▷ TensorBimod.X Q L)
                            (Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ 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)
                                (TensorBimod.X Q L))
                              (P.actRight ▷ TensorBimod.X Q L)
                              (Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ P.X ◁ TensorBimod.actLeft Q L)))
                    ⋯))
                ⋯ ▷
              U.X :
            tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X (P.tensorBimod Q) L) U.X ⟶
              tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X P (Q.tensorBimod L)) U.X)
          (TensorBimod.actRight P (Q.tensorBimod L) :
            tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X P (Q.tensorBimod L)) U.X ⟶
              TensorBimod.X P (Q.tensorBimod L))) :
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorRight U.X) (tensorObj.{v₁, u₁} (𝒞 := inst) (P.tensorBimod Q).X L.X) ⟶
        TensorBimod.X P (Q.tensorBimod L))

Tactic:
slice_lhs 3 4 => rw [coequalizer.π_desc]

Premises used: CategoryTheory.Limits.coequalizer.π_desc