Exercise 1.5.2.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $K \subseteq \Delta ^1 \times \Delta ^1$ be the simplicial subset appearing in Example 1.5.2.8. Suppose we are given a diagram $\sigma : K \rightarrow \operatorname{\mathcal{C}}$, which we depict graphically as
Composing with the unit map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$, we obtain a diagram $\sigma '$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, which we can depict as
Show that the diagram $\sigma '$ is commutative if and only if $\sigma $ can be extended to a map $\overline{\sigma }: \Delta ^1 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$. Beware that this extension is generally not unique.