At the page claims 29 of HTT, Lurie claims that the category $hS$ (we ignore all the enrichement) admits the following presentation by generators and relation :
The objects of $hS$ are the vertices of $S$ (this one is clear).
For every edge $ f: \Delta^1 \rightarrow S$ there is a morphism $\overline{f}$ in $hS$.
For each $2$-simplex $\sigma$ we have $\overline{d_0\sigma} \circ \overline{d_2\sigma} = \overline{d_1\sigma}$.
For each vertex $x$ the morphism $\overline{s_0x}$ is the identity $id_x$.
He claims that this follows from the fact the functor $h$ is left adjoint to the nerve functor but I don't see how.
We first idea was to use Yoneda and try to use the adjunction on $sSet( \Delta^1, S)$ to get the statement about morphisms but the adjunction isnot in the correct direction. Then I tought about using that $h\Delta^1 \cong [1]$ as a category and use that that the set of morphisms in $hS$ is in bijection with $Cat([1], hS)$ but I did not get much result.