Remark 2.4.4.12. Let $u: S_{\bullet } \hookrightarrow S'_{\bullet }$ be a monomorphism of simplicial sets. Then, for each $m \geq 0$, $u$ induces a monomorphism of sets $E(S,m) \hookrightarrow E(S',m)$ (see Notation 2.4.4.9). It follows from Theorem 2.4.4.10 that if $x$ and $y$ are vertices of $S_{\bullet }$, then the induced map of simplicial sets $\operatorname{Hom}_{ \operatorname{Path}[S]}(x,y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{Path}[S'] }( u(x), u(y) )_{\bullet }$ is a monomorphism.
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