nLab
product-preserving functor

Let $F:C\to D$ be a functor, and suppose a collection of objects $\{c_i\}$ in $C$ admits a product, with projections

$${\pi }_i:\prod _i{c}_i\to c_i.$$\pi_i \colon \prod_i c_i \to c_i.

We say $F$ preserves this product if the collection of maps

$$F({\pi }_i):F(\prod _i{c}_i)\to F(c_i)$$F(\pi_i): F(\prod_i c_i) \to F(c_i)

exhibits $F({\prod }_i{c}_i)$ as a product of the collection of objects $F(c_i)$.

If $C$ has all (small) products, $F$ is product-preserving if it preserves every product in $C$.

If $C$ does not have all small products, then one wants a more subtle condition; compare flat functor (which is about finite limits instead of products).