Remark 10.2.2.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. It follows from Proposition 10.2.2.14 that the diagram of $\infty $-categories
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj},+}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \ar [r] & \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}}), \operatorname{\mathcal{C}})} \]
is a categorical pullback square. In particular, if $X_{\bullet }$ is a simplicial object of $\operatorname{\mathcal{C}}$, then the datum of an augmentation of $X_{\bullet }$ is equivalent to the datum of an augmentation on the underlying semisimplicial object of $X_{\bullet }$.