Let $\mathcal{C}$ be a category and $h_-: \mathcal{C} \to \mathrm{Set}^{\mathcal{C}^{op}}$ be the Yoneda embedding. Let $\mathcal{A}$ be a cocomplete category and $F: \mathcal{C} \to \mathcal{A}$ a functor. Denote $G$ the left Kan extension of $F$. Then, is $G$ continuous, i.e. does it commute with colimits?
Does this extend to the world of $\infty$-categories? Namely, let $\mathcal{C}$ be an $\infty$-category and $h_-: \mathcal{C} \to (\infty-\mathrm{groupoids})^{\mathcal{C}^{op}}$ be the Yoneda embbeding. Let $\mathcal{A}$ be a cocomplete $\infty$-category, $F: \mathcal{C}\to \mathcal{A}$ a functor and $G$ its left Kan extension. Then, we can also ask if $G$ is continuous. Here, colimit is of course understood to be in the appropriate sense, i.e. homotopy colimit.
Thank you.