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 $$ 0\to\mathbb{Z_X}\to\mathcal{C}^0\to\mathcal{C}^1\to\ldots $$ 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"?