Functor

Multifunctor

A multifunctor is a functor from the product category . cat This is stronger to a mapping on objects and morphisms which is functorial in each argument when all other arguments are held constant, viewing objects as identities.

Counterexample

Let be groups-as-categories. Then is the direct product of groups, and a bifunctor is a group action of on an object of . Functoriality in both arguments, on the other hand, makes a group action of the free product of groups on an object of .

In fact, if is a mapping functorial in each argument, namely and are functors for any and , then is a bifunctor iff the following diagram commutes for any and :

This essentially says the two parts of a bifunctor commute.

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