-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
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
refine :
  Eq (α :=
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X))
        (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X) ⟶
      TensorBimod.X (P.tensorBimod Q) L)
    (CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft (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)))
      (CategoryTheory.Functor.map (tensorLeft (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X))
        (coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)))
      (P.actRight ▷ TensorBimod.X Q L ≫
        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)))
          (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) ≫
              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))
            ⋯)))
    (CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft (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)))
      (CategoryTheory.Functor.map (tensorLeft (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X))
          (coequalizer.π (Q.actRight ▷ L.X) (Iso.hom (α_ Q.X T.X L.X) ≫ Q.X ◁ L.actLeft)) :
        Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X))
            (tensorObj.{v₁, u₁} (𝒞 := inst) Q.X L.X) ⟶
          Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft (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)))
      ((Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ P.X ◁ TensorBimod.actLeft Q L) ≫
          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)))
            (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) ≫
                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))
              ⋯) :
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L) ⟶
          TensorBimod.X (P.tensorBimod Q) L))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L) ⟶
      TensorBimod.X (P.tensorBimod Q) L)
    (P.actRight ▷ TensorBimod.X Q L ≫
        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)))
          (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) ≫
              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))
            ⋯) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L) ⟶
        TensorBimod.X (P.tensorBimod Q) L)
    ((Iso.hom (α_ P.X S.X (TensorBimod.X Q L)) ≫ P.X ◁ TensorBimod.actLeft Q L) ≫
        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)))
          (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) ≫
              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))
            ⋯) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) P.X S.X) (TensorBimod.X Q L) ⟶
        TensorBimod.X (P.tensorBimod Q) L)

Tactic:
refine (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 ?_

Premises used: CategoryTheory.Functor.map, CategoryTheory.MonoidalCategory.tensorLeft, Iff.mp, CategoryTheory.Limits.coequalizer.π, CategoryTheory.cancel_epi