Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 4.3.2.10. Let $\operatorname{\mathcal{C}}$ be a category and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor between categories. For every functor $U: \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ extending $G$, let $\overline{F}(U)$ denote the composite functor

\[ \operatorname{\mathcal{C}}\xrightarrow { \overline{\iota }_{\operatorname{\mathcal{C}}} } ( \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}})_{ / \iota _{\operatorname{\mathcal{D}}} } \xrightarrow {U } \operatorname{\mathcal{E}}_{ / (U \circ \iota _{\operatorname{\mathcal{D}}} ) } = \operatorname{\mathcal{E}}_{/G}. \]

Then the construction $U \mapsto \overline{F}(U)$ induces a bijection

\[ \{ \textnormal{Functors $U: \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ satisfying $U|_{\operatorname{\mathcal{D}}} = G$} \} \rightarrow \{ \textnormal{Functors $\overline{F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}_{/G}$} \} . \]