nLab product-preserving functor

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

$$\pi_i \colon \prod_i c_i \to c_i.$$

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

$$F(\pi_i): F(\prod_i c_i) \to F(c_i)$$

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).