Let $\mathcal{C}$ be the product of categories $\mathcal{C}_1,\ldots,\mathcal{C}_n$ and $\mathcal{D}$ be any category. A multifunctor $F:\mathcal{C}\to \mathcal{D}$ satisfies the following

• for every $A\in\operatorname{Ob}(\mathcal{C})$ , $F(A)\in \operatorname{Ob}(\mathcal{D})$ ,
• for every $\alpha\in \operatorname{Mor}(\mathcal{C})$ , $F(\alpha)\in \operatorname{Mor}(\mathcal{D}): F(A)\to F(B)$ , for some objects $A,B\in \operatorname{Ob}(\mathcal{C})$ ,
• There is a function $\phi: \lbrace 1,\ldots, n\rbrace\to \lbrace \operatorname{id},\operatorname{op}\rbrace$ where $\operatorname{id}$ is the identity functor and $\operatorname{op}$ is the opposite functor, with $$\overline{C}:= \phi(1)(\mathcal{C}_1) \times \cdots \times \phi(n)(\mathcal{C}_n)\qquad\mbox{and}\qquad \overline{\alpha_i}:=\phi(i)(\alpha_i)$$ for any morphisms $\alpha_i$ in $\mathcal{C}_i$ , such that $\overline{F}: \overline{\mathcal{C}}\to \mathcal{D}$ given by $$\overline{F}(A):=F(A)\qquad\mbox{and}\qquad \overline{F}(\alpha_1,\ldots,\alpha_n):=F(\overline{\alpha_1},\ldots, \overline{\alpha_n})$$ is a covariant functor.

Remarks.

1. The function $\phi$ in condition 3 above can be changed so that $\overline{F}$ is a contravariant functor instead.
2. Since $\overline{F}$ is a functor, by restricting $\overline{F}$ to any coordinate gives us a functor as well. Formally, given object $A=(A_1,\ldots,A_n) \in \overline{\mathcal{C}}$ , if we define $F_A:\phi(i)(\mathcal{C}_i)\to D$ by setting $$F_A(B_i)= \overline{F}(\hat{A_i}(B_i))\qquad \mbox{and}\qquad F_A(\beta_i)= \overline{F}(\hat{A_i}(\beta_i)),$$ for each $i$ , where
   1. $\hat{A_i}(B_i)$ is the object in $\overline{\mathcal{C}}$ whose $i$ -th coordinate is $B_i$ and agrees with $A$ everywhere else, and
   2. $\hat{A_i}(\beta_i)$ is the morphism in $\overline{\mathcal{C}}$ whose $i$ -th coordinate is $\beta_i$ , and the identity morphism (on $A_j$ ) everywhere else,
   then $F_A$ is a covariant functor.
3. Furthermore, since $\overline{F}$ is covariant, this means for any $\alpha=(\alpha_1,\ldots, \alpha_n): A\to B$ , we have the decomposition $$\overline{F}(\alpha)= \overline{F}(\hat{\alpha_1})\circ \cdots \circ \overline{F}(\hat{\alpha_n})$$ where $\hat{\alpha_i}:=\hat{A_i}(\alpha_i)$ as defined in property 2 above.
4. In addition, we see that $\overline{F}(\hat{\alpha_i})\circ \overline{F}(\hat{\alpha_j}) = \overline{F}(\hat{\alpha_j}) \circ \overline{F}(\hat{\alpha_i})$ for $i\ne j$ .
5. In fact, properties 2, 3, and 4 are enough to insure that $\overline{F}$ is a covariant functor, for \begin{eqnarray*} \overline{F}(\alpha\circ \beta) &=& \overline{F}((\alpha_1,\ldots,\alpha_n)\circ (\alpha_n,\ldots, \beta_n)) \\ &=& \overline{F}(\alpha_1\circ \beta_1,\ldots, \alpha_n\circ \beta_n) \\ &=& \overline{F}(\widehat{\alpha_1\circ \beta_1})\circ \cdots \circ \overline{F}(\widehat{\alpha_n\circ \beta_n}) \\ &=& F_A(\alpha_1\circ \beta_1)\circ \cdots \circ F_A(\alpha_n\circ \beta_n) \\ &=& (F_A(\alpha_1)\circ F_A(\beta_1))\circ \cdots \circ (F_A(\alpha_n)\circ F_A(\beta_n)) \\ &=& (\overline{F}(\hat{\alpha_1})\circ \overline{F}(\hat{\beta_1}))\circ \cdots \circ (\overline{F}(\hat{\alpha_n})\circ \overline{F}(\hat{\beta_n})) \\ &=& (\overline{F}(\hat{\beta_1})\circ \overline{F}(\hat{\alpha_1}))\circ \cdots \circ (\overline{F}(\hat{\beta_n})\circ \overline{F}(\hat{\alpha_n})) \\ &=& \overline{F}(\widehat{\beta_1\circ \alpha_1})\circ \cdots \circ \overline{F}(\widehat{\beta_n\circ \alpha_n}) \\ &=& \overline{F}(\beta_1\circ \alpha_1,\ldots, \beta_n\circ \alpha_n) \\ &=& \overline{F}(\beta\circ \alpha) \end{eqnarray*}This means we can replace the statement that $\overline{F}$ is a covariant functor in condition 3 of the definition by the three properties above.
6. $F$ is called a bifunctor or trifunctor whenever $n=2$ or $3$ .

Hom functors. The most famous bifunctor is the $\hom$ functor from $\mathcal{C}\times \mathcal{C}\to {Set}$ . Given objects $A,B$ in $\mathcal{C}$ , $\hom(A,B)$ is the set of all morphisms from $A$ to $B$ . In addition, given morphisms $\alpha:A\to B$ and $\beta:X\to Y$ , $\hom(\alpha,\beta)$ is the morphism from $\hom(B,X)$ $\hom(A,Y)$ taking $f:B\to X$ to $g:=\beta \circ f \circ \alpha:A\to Y$ :

Bibliography

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A. J. Berrick, M. E. Keating, Categories and Modules, with K-theory in View, Cambridge University Press (2000).