Product-preserving functor

This article defines a functor property: a property that can be evaluated to true/false given a functor between two categories.
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Definition

Symbol-free definition

A functor between two categories is termed product-preserving if it preserves categorical products.

Definition with symbols

Suppose $\mathcal{C},\mathcal{D}$ are categories. A functor $\mathcal{F}:\mathcal{C}\to \mathcal{D}$ is termed product-preserving if the following holds. Suppose $A_1,A_2\in \mathrm{O}\mathrm{b}\mathcal{C}$ and $C\in \mathrm{O}\mathrm{b}\mathcal{C}$ is a product of $A$ and $B$ with projection maps ${\pi }_i:C\to A_i$.

Then, $\mathcal{F}C$, along with the projection maps $\mathcal{F}{\pi }_1,\mathcal{F}{\pi }_2$, is a product of $\mathcal{F}{A}_1$ and $\mathcal{F}{A}_2$.

Note that this term is typically used for categories that both admit finite products, though it makes sense for arbitrary categories.