Kerodon

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

Remark 8.5.3.9. The category $\operatorname{Ret}$ of Construction 8.5.0.1 contains an initial object $\widetilde{Y}$. It follows that the inclusion map $\operatorname{Idem}\hookrightarrow \operatorname{Ret}$ has a unique extension $T: \operatorname{Idem}^{\triangleleft } \rightarrow \operatorname{Ret}$ carrying the cone point of $\operatorname{Idem}^{\triangleleft }$ to the object $\widetilde{Y}$. Unwinding the definitions, we see that a functor of $\infty $-categories $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$ is right Kan extended from $\operatorname{N}_{\bullet }( \operatorname{Idem})$ if and only if the composition

\[ \operatorname{N}_{\bullet }( \operatorname{Idem})^{\triangleleft } \xrightarrow { \operatorname{N}_{\bullet }(T) } \operatorname{N}_{\bullet }( \operatorname{Ret}) \xrightarrow { F } \operatorname{\mathcal{C}} \]

is a limit diagram. Proposition 8.5.3.8 asserts that this condition is automatically satisfied. In particular, the object $\overline{F}( \widetilde{Y} )$ is a limit of the underlying diagram $F = \overline{F}|_{ \operatorname{N}_{\bullet }( \operatorname{Idem})}$. Similarly, $\overline{F}( \widetilde{Y} )$ is a colimit of the diagram $F$.