This article is about functors of two variables. Possibly the term ‘bifunctor’ has been used for a functor between bicategories (citation?), but such usage (if it exists) seems to be rare; the usual term for that is ‘pseudo functor’.^{1}
A bifunctor (short for binary functor, that is $2$-ary) or functor of two variables is simply a functor whose domain is the product of two categories.
For for $C_1$, $C_2$ and $D$ categories, a functor
is also called a bifunctor from $C_1$ and $C_2$ to $D$.
Famous bifunctors are
the hom functor
on any small category $C$, or if $C$ is a closed category, the internal hom functor
on every monoidal category $(C, \otimes)$ the tensor product functor
bifunctor, two-variable adjunction, Quillen bifunctor
Outside of certain computer science contexts, it is not clear that the term ‘bifunctor’ is used frequently nowadays, even for the sense of a functor of two variables. It is used more frequently in older texts. ↩