This the definition of a Quillen bifunctor from nlab.
Let $C,D,E$ be model categories. $$ \otimes: C \times D \rightarrow E$$ is a Quillen bifunctor if it preserves colimits in each variable and
Condition: Given $f:a \rightarrow b, g:c \rightarrow d$, both cofibrations, the canonical pushout map $$ a \otimes d\sqcup_{a \otimes c} b \otimes c \rightarrow b \otimes d$$ is a cofibration. And is trivial if either one of $f,g$ is trivial.
It is this condition that I don't know why it is justified. If one were to define it, wouldn't the natural condition be:
Condition': If $\otimes$ is a Quillen functor when $C \times D$ is equipped with the product model structure, i.e. a morphism is cof,fib,we, iff each component is.
Why is this not taken as the definition?