-- 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
h :
  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)
        (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)))
            ⋯)) =
    P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ≫
      coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X)
          (Iso.hom (α_ P.X S.X (Q.tensorBimod L).X) ≫ P.X ◁ (Q.tensorBimod L).actLeft) ≫
        𝟙 (TensorBimod.X P (Q.tensorBimod L))
⊢ CategoryStruct.comp (Y := TensorBimod.X (P.tensorBimod Q) L)
      (Iso.inv (α_ P.X Q.X L.X) ≫
        coequalizer.π (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_ (TensorBimod.X P Q) T.X L.X) ≫ TensorBimod.X P Q ◁ 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)))
            ⋯))
        ⋯) =
    P.X ◁ coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft) ≫
      coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X)
          (Iso.hom (α_ P.X S.X (Q.tensorBimod L).X) ≫ P.X ◁ (Q.tensorBimod L).actLeft) ≫
        𝟙 (TensorBimod.X P (Q.tensorBimod L))

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

Premises used: CategoryTheory.Limits.coequalizer.π_desc