Let $X$ be a topological space, and $\mathcal{F}^\bullet$ be a cochain complex of sheaves on $X$. The hypercohomology $\mathbb{H}^i(X,\,\mathcal{F}^\bullet)$ is defined as $$\mathbb{H}^i(X,\,\mathcal{F}^\bullet)=\mathrm{H}^i[\mathrm{Tot}^{\prod}\Gamma(I^{\bullet\bullet})]\,,$$ where $I^{\bullet\bullet}$ is a(ny) Cartan-Eilenberg resolution of $\mathcal{F}^\bullet$ and $\mathrm{Tot}^{\prod}$ means the direct product totalization.
My question is that are there any particular reasons why we use the direct product totalization instead of the direct sum totalization? Thanks a lot.