Remark 5.1.4.7 (Products). Let $\{ q_ i: X_ i \rightarrow S_ i \} _{i \in I}$ be a collection of cartesian fibrations indexed by a set $I$. Then the product map $q: \prod _{i \in I} X_ i \rightarrow \prod _{i \in I} S_ i$ is also a cartesian fibration. Moreover, an edge of $X = \prod _{i \in I} X_ i$ is $q$-cartesian if and only if its image in each $X_ i$ is $q_ i$-cartesian. See Remark 5.1.1.7
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