sheaf theory - Descent of singular cohomology

When proving that singular cohomology of an appropriate space $X$ equals sheaf cohomology of $X$ with "values" (does one say that?) in the sheaf $mathbb{Z}_X$ of locally constant functions, the author of the proof I have read observes that the sheafifications $mathcal{C}^n$ of the singular cochain complexes $C^n(-)=hom_{Ab}(mathbb{Z}hom_{Top}(Delta^n,-),mathbb{Z})$ form an injective resolution
$$
0tomathbb{Z_X}tomathcal{C}^0tomathcal{C}^1toldots
$$
of $mathbb{Z}_X$.

Why must one sheafify the singular cochain complexes? Aren't they sheaves since they satisfy descent (= have the excision property) and "sheaf=presheaf+descent"?