Lemma 2.2.11 in mn.uio.no/math/personer/vit/rognes/papers/aoms186-nocrop.pdf shows that the surjection you denote sd(X) --> 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.
John Rognes
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