$\newcommand{\Mod}{\operatorname{Mod}}$
Let $A, B$ be two rings, and let $F:\Mod A\times \Mod A \to \Mod B$ be a biadditive bifunctor.
I want to extend $F$ naturally to a bifunctor from complexes over $A$ to complexes over $B$.
There are two ways I know how do to it, one is direct sum totalization, setting
$$F^\oplus (M,N)^n = \bigoplus_{p+q=n} F(M^p,N^q)$$
The other one is direct product totalization,
$$F^\prod (M,N)^n = \prod_{p+q=n} F(M^p,N^q).$$
For example, if $A$ is commutative, and $F$ is tensor product over $A$, one uses direct sum totalization to get the tensor product of complexes.
If $A$ is commutative and $F$ is the $\operatorname{Hom}_A(-,-)$ bifunctor, then one uses direct product totalization to get the Hom between two complexes.
My question - how does one knows which of these to use, in order to get the "correct" extension? why do we use direct sum for tensor and direct product for hom? and what to do in general for other biadditive bifunctors?