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 $S(X)$.

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 $S(X)$. In the book above, we called this the desingularization of $sd(X)$.

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.

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 $S(X)$.