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Lemma 2.2.11 in mn.uio.no/math/personer/vit/rognes/papers/aoms186-nocrop.pdf shows that the surjection you denote $sd(X) \to S(X)$ is an isomorphism if and only if $X$ is a non-singular simplicial set, i.e., if and only if each non-degenerate simplex in $X$ is embedded. It may be interesting to determine for which $X$ the initial non-singular simplicial set under $sd(X)$ is precisely $S(X)$.