Example 10.2.5.19. Let $C_{\bullet }$ be an augmented semisimplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$. For small values of $k$, the face cube $\tau _ k: \operatorname{\raise {0.1ex}{\square }}^{k+1} \rightarrow \operatorname{\mathcal{C}}$ of Construction 10.2.5.18 can be described more explicitly:
The face cube $\tau _{-1}$ can be identified with the object $C_{-1}$ of $\operatorname{\mathcal{C}}$.
The face cube $\tau _0$ can be identified with the face operator $d^{0}_{0}: C_0 \rightarrow C_{-1}$.
The face cube $\tau _1$ is a square diagram
\[ \xymatrix@R =50pt@C=50pt{ C_1 \ar [r]^{ d^{1}_{0} } \ar [d]^{ d^{1}_{1} } & C_0 \ar [d]^{ d^{0}_{0} } \\ C_0 \ar [r]^{ d^{0}_{0} } & C_{-1} } \]which witnesses the identity $[ d^{0}_{0} ] \circ [d^{1}_{0} ] = [ d^{0}_{0} ] \circ [d^{1}_{1} ]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.