-- 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
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
| (CategoryStruct.comp (Y :=
      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))
      (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)))
            ⋯))
        ⋯) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (P.tensorBimod Q).X L.X ⟶ TensorBimod.X P (Q.tensorBimod L))
have a : (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)))
      sorry) :
  tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X P Q) L.X ⟶ TensorBimod.X P (Q.tensorBimod L))
Tactic: rw [coequalizer.π_desc]