-- 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
dsimp :
  Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X (P.tensorBimod Q) L) U.X ⟶ TensorBimod.X P (Q.tensorBimod L))
    (TensorBimod.actRight (P.tensorBimod Q) L ≫
        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) (TensorBimod.X (P.tensorBimod Q) L) U.X ⟶ TensorBimod.X P (Q.tensorBimod L))
    (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)) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X (P.tensorBimod Q) L) U.X ⟶ TensorBimod.X P (Q.tensorBimod L))
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (TensorBimod.X (P.tensorBimod Q) L) U.X ⟶ TensorBimod.X P (Q.tensorBimod L))
    (TensorBimod.actRight (P.tensorBimod Q) L ≫ hom P Q L) (hom P Q L ▷ U.X ≫ TensorBimod.actRight P (Q.tensorBimod L))

Tactic:
dsimp [hom, homAux]

Premises used: Bimod.AssociatorBimod.hom, Bimod.AssociatorBimod.homAux