Lemma 2.2.11 in [mn.uio.no/math/personer/vit/rognes/papers/aoms186-nocrop.pdf][1] 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.


  [1]: https://mn.uio.no/math/personer/vit/rognes/papers/aoms186-nocrop.pdf