Remark 11.4.0.7. The universal mapping simplex $\operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }})$ is equipped with a morphism of simplicial sets $\pi : \operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }})$, given on $n$-simplices by the formula $\pi ( \overrightarrow {X}, \sigma ) = \overrightarrow {X}$. We will refer to $\pi $ as the projection map. Unwinding the definitions, we see that for every simplicial set $K$, there is a canonical isomorphism of simplicial sets
\[ K \simeq \{ K\} \times _{ \operatorname{N}_{\bullet }(\operatorname{Set_{\Delta }}) } \operatorname{N}^{+}_{\bullet }(\operatorname{Set_{\Delta }}). \]