Example 7.6.5.4. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, and suppose we are given a collection of objects $\{ X(n) \} _{n \geq 0}$ and morphisms $f_ n: X(n) \rightarrow X(n+1)$ in $\operatorname{\mathcal{C}}$. It follows from Remark 7.6.5.3 that the diagram
can be extended to a functor $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$. In fact, there is a preferred choice of such an extension, which is uniquely determined by the requirement that it factors through the inclusion map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$.