Let $p: \mathscr{C}\to\mathscr{D}$ be a functor of (small) $\infty$-categories. Let $\mathscr{E}$ be a cocomplete $\infty$-category. Assume that $\mathscr{C}, \mathscr{D}, \mathscr{E}$ admit finite products and $p$ preserves finite products. Denote ${\rm Fun}^{\times}(\mathscr{D}, \mathscr{E})$ the $\infty$-category of those functors preserving finite products. I want to know if the following result holds:
The left Kan extension along $p: \mathscr{C}\to\mathscr{D}$ induces a functor ${\rm Fun}^{\times}(\mathscr{C}, \mathscr{E})\to{\rm Fun}^{\times}(\mathscr{D}, \mathscr{E})$, which is a cocartesian fibration of $\infty$-categories if $p$ is a cartesian fibration.
Could anyone give an exposition on this? Hopefully, it is in Lurie's HTT/HA/Kerodon/DAG etc., or follows easily from there, but I can't find it. In my situation, $p$ is the nerve of a cartesian fibration of ordinary categories, I hope it holds in the above more general form. Adding other mild conditions (e.g. $\mathscr{E}$ is presentable) would be fine also.
