Construction 3.3.1.6. Let $S_{\bullet }$ be a semisimplicial set. For each $n \geq 0$, we let $S^{+}_{n}$ denote the collection of pairs $( \alpha , \tau )$ where $\alpha : [n] \twoheadrightarrow [m]$ is a nondecreasing surjection of linearly ordered sets and $\tau $ is an element of $S_{m}$.
Let $\beta : [n'] \rightarrow [n]$ be a morphism in the category $\operatorname{{\bf \Delta }}$. For every element $(\alpha , \tau ) \in S^{+}_{n}$, the composite map $\alpha \circ \beta : [n'] \rightarrow [m]$ factors uniquely as a composition $[n'] \xrightarrow {\alpha '} [m'] \xrightarrow {\beta '} [m]$, where $\alpha '$ is surjective and $\beta '$ is injective. We define a map $\beta ^{\ast }: S^{+}_{n} \rightarrow S^{+}_{n'}$ by the formula $\beta ^{\ast }( \alpha , \tau ) = ( \alpha ', \beta '^{\ast }(\tau ) ) \in S_{n'}^{+}$.