Kerodon

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

Corollary 1.1.4.7. Let $\operatorname{\mathcal{C}}$ be a category. Then the evaluation functor

\[ \operatorname{ev}_{0}: \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}\quad \quad X_{\bullet } \mapsto X_0 \]

admits a left adjoint, given on objects by the formation of constant simplicial objects $C \mapsto \underline{C}_{\bullet }$ described in Construction 1.1.4.2.