Kerodon

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Remark 5.7.6.5. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. Suppose we are given an object $Y \in \operatorname{\mathcal{D}}$ and a vertex $\eta \in \mathscr {F}( U(Y) )$. Then:

  • If $U$ is fully faithful and $\eta $ exhibits the functor $\mathscr {F}$ as corepresented by $U(Y)$, then it also exhibits the functor $\mathscr {F} \circ U$ as corepresented by $\eta $.

  • If $U$ is an equivalence of $\infty $-categories and $\eta $ exhibits the functor $\mathscr {F} \circ U$ as corepresented by $\eta $, then it also exhibits $\mathscr {F}$ as corepresented by $U(Y)$.

  • If $U$ is an equivalence of $\infty $-categories, then the functor $\mathscr {F}$ is corepresentable if and only if $\mathscr {F} \circ U$ is corepresentable.