Remark 7.6.1.21. In the situation of Proposition 7.6.1.20, let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which preserves the limits of each of the diagrams $f_ i$. Suppose that the collection $\{ X_ i \} _{i \in I}$ admits a product in $\operatorname{\mathcal{C}}$. Then the product of $\{ X_ i \} _{i \in I}$ is preserved by the functor $F$ if and only if the limit of $f$ is preserved by the functor $F$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$