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Remark The inclusion functor $\operatorname{{\bf \Delta }}\hookrightarrow \operatorname{Lin}_{\neq \emptyset }$ has a unique left inverse $R: \operatorname{Lin}_{\neq \emptyset } \rightarrow \operatorname{{\bf \Delta }}$, given on objects by the formula $R(I) = [n]$ when $I$ has cardinality $n+1$. It follows that the equivalence $\operatorname{Fun}_{\ast }( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set}) \rightarrow \operatorname{Set_{\Delta }}$ of Proposition admits an explicit right inverse, which carries a simplicial set $X: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ to the functor $X^{+}: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$ given by the formula

\[ X^{+}(I) = \begin{cases} X( R(I) ) & \textnormal{ if $I$ is nonempty } \\ \ast & \textnormal{ otherwise. } \end{cases} \]