[[Functor]]
# Multifunctor

A **multifunctor** $F : \prod_{\alpha \in A} \cat C_{\alpha} \to \cat D$ is a [[functor]] from the [[product category]] $\prod_{\alpha \in A} \cat C_{\alpha}$. #m/def/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]].

> [!tip]+ Counterexample
> Let $G,H$ be [[Groups as categories|groups-as-categories]].
> Then $G \times H$ is the [[direct product of groups]],
> and a bifunctor $F : G \times H \to \cat C$ is a [[group action]] of $G \times H$ on an object of $\cat C$.
> Functoriality in both arguments, on the other hand, makes $F$ a group action of the [[free product of groups]] on an object of $\cat C$. <span class="QED"/>

In fact, if $F : \cat C \times \cat D \to \cat E$ is a mapping functorial in each argument, namely $F(C,-)$ and $F(-,D)$ are functors for any $C \in \Ob \cat C$ and $D \in \cat D$,
then $F$ is a bifunctor iff the following diagram commutes for any $c \in \cat C(C,C')$ and $d \in \cat D(D,D')$:

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This essentially says the two parts of a bifunctor _commute_.

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