= Bifunctor
{title2=$F:\mathcal C\times\mathcal D\to\mathcal E$}
A <bifunctor> is a <functor> on the product of two <categories>. It assigns $F(c,d)$ to each pair of objects and $F(u,v)$ to a pair of <morphisms>, preserving <identity morphisms> and <composition in a category>. Equivalently, it is functorial separately in each argument, with the two actions commuting: $F(1,v)F(u,1)=F(u,1)F(1,v)$ whenever the expressions have the corresponding sources and targets. The product <category> has pairs as objects, pairs of <morphisms> as arrows, and componentwise <composition in a category>. A <monoidal tensor product> is a <bifunctor> from a <category> times itself.
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