Multifunctor

A multifunctor $𝐹 : \prod 𝛼 \in 𝐴 𝖢 𝛼 \to 𝖣$ is a functor from the product category $\prod 𝛼 \in 𝐴 𝖢 𝛼$ . 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 $𝐺, 𝐻$ be groups-as-categories. Then $𝐺 \times 𝐻$ is the direct product of groups, and a bifunctor $𝐹 : 𝐺 \times 𝐻 \to 𝖢$ is a group action of $𝐺 \times 𝐻$ on an object of $𝖢$ . Functoriality in both arguments, on the other hand, makes $𝐹$ a group action of the free product of groups on an object of $𝖢$ .

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

This essentially says the two parts of a bifunctor commute.

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Multifunctor

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