# Kerodon

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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

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