-- Module: Mathlib.CategoryTheory.Shift.ShiftedHomOpposite
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
X Y Z : C
n m : ℤ
f : ShiftedHom X Y n
g : ShiftedHom Y Z m
q : ℤ
hq : n + m = q
dsimp :
  CategoryStruct.comp (Y :=
        Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n)
          (Opposite.unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ n) (Opposite.op X))))
        (Quiver.Hom.unop
            (NatTrans.app (Iso.inv (Equivalence.unitIso (opShiftFunctorEquivalence C n))) (Opposite.op X)) :
          Opposite.unop (Opposite.op X) ⟶
            Opposite.unop
              (Opposite.op
                (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n)
                  (Opposite.unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ n) (Opposite.op X))))))
        (Functor.map (shiftFunctor C n)
            (CategoryStruct.comp (Y :=
                Opposite.unop
                  (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ n)
                    (Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) Y))))
                (Quiver.Hom.unop (Functor.map (shiftFunctor Cᵒᵖ n) (Quiver.Hom.op f)) :
                  Opposite.unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ n) (Opposite.op X)) ⟶
                    Opposite.unop
                      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ n)
                        (Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) Y))))
                (Quiver.Hom.unop
                    (NatTrans.app (Iso.inv (Equivalence.counitIso (opShiftFunctorEquivalence C n))) (Opposite.op Y)) :
                  Opposite.unop
                      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ n)
                        (Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) Y))) ⟶
                    Opposite.unop (Opposite.op Y)) ≫
              g) :
          Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n)
              (Opposite.unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ n) (Opposite.op X))) ⟶
            Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n)
              (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C m) Z)) ≫
      NatTrans.app (Iso.inv (shiftFunctorAdd' C m n q ⋯)) Z =
    f ≫ Functor.map (shiftFunctor C n) g ≫ NatTrans.app (Iso.inv (shiftFunctorAdd' C m n q ⋯)) Z
⊢ CategoryStruct.comp (Y :=
        Opposite.unop
          (Functor.obj.{v_1, v_1, u_1, u_1}
            (Equivalence.functor (opShiftFunctorEquivalence C n) ⋙ Equivalence.inverse (opShiftFunctorEquivalence C n))
            (Opposite.op X)))
        (Quiver.Hom.unop (NatTrans.app (Iso.inv (Equivalence.unitIso (opShiftFunctorEquivalence C n))) (Opposite.op X)))
        (Functor.map (shiftFunctor C n)
          (Quiver.Hom.unop
            (Quiver.Hom.op g ≫
              CategoryStruct.comp (Y :=
                Functor.obj.{v_1, v_1, u_1, u_1}
                  (Equivalence.inverse (opShiftFunctorEquivalence C n) ⋙
                    Equivalence.functor (opShiftFunctorEquivalence C n))
                  (Opposite.op Y))
                (NatTrans.app (Iso.inv (Equivalence.counitIso (opShiftFunctorEquivalence C n))) (Opposite.op Y))
                (Functor.map (shiftFunctor Cᵒᵖ n) (Quiver.Hom.op f))))) ≫
      NatTrans.app (Iso.inv (shiftFunctorAdd' C m n q ⋯)) Z =
    f.comp g ⋯

Tactic:
dsimp [comp]

Premises used: CategoryTheory.ShiftedHom.comp