$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark In the situation of Corollary, one can think of the simplicial set

\[ Z_{\bullet } = \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \underset {\operatorname{Fun}( \Lambda ^2_1, \operatorname{\mathcal{C}}) }{\times } \{ (g, \bullet , f) \} \]

as a “parameter space” for all choices of $2$-simplex $\sigma $ satisfying $d_0(\sigma ) = g$ and $d_2(\sigma ) = f$ (note that such $2$-simplices can be identified with the vertices of $Z_{\bullet }$). Consequently, we can summarize Corollary informally by saying that this parameter space is contractible.