Corollary 4.3.2.16. Let $\operatorname{\mathcal{D}}$ be a category and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ be a functor between categories. For every functor $U: \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ extending $F$, let $\overline{G}(U)$ denote the composite functor
\[ \operatorname{\mathcal{D}}\xrightarrow { \overline{\iota }_{\operatorname{\mathcal{D}}} } ( \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}})_{ \iota _{\operatorname{\mathcal{C}}} / } \xrightarrow {U } \operatorname{\mathcal{E}}_{ (U \circ \iota _{\operatorname{\mathcal{C}}} )/ } = \operatorname{\mathcal{E}}_{F/}. \]
Then the construction $U \mapsto \overline{G}(U)$ induces a bijection
\[ \{ \textnormal{Functors $U: \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ satisfying $U|_{\operatorname{\mathcal{C}}} = F$} \} \rightarrow \{ \textnormal{Functors $\overline{G}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}_{F/}$} \} . \]