Corollary 10.2.5.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks. Then the forgetful functor
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(\Delta ^1,\operatorname{\mathcal{C}}) \quad \quad C_{\bullet } \mapsto (d^{0}_{0}: C_0 \rightarrow C_{-1}) \]
admits a right adjoint, given on objects by the construction $(f: X \rightarrow Y) \mapsto \operatorname{\check{C}}_{\bullet }(X/Y)$.