Let $y:\Delta^0\to S$ be a vertex of a quasicategory $S$. According to the proof of HTT Theorem 2.2.4.1, we can show that $St_S y (x)\cong Map_{\mathfrak{C}[S]}(x,y)$ for any vertex $x$ of $S$. How can we show this? We can see by definition that $St_S y (x):= Map_M(x,v)$ where $M$ is the simplicial category given by the pushout

$\mathfrak{C}[(\Delta^0)^\triangleright]\coprod_{\mathfrak{C}[\Delta^0]} \mathfrak{C}[S],$and $v$ is the image of the cone point. Also, what is the natural map that we have

$(\Delta^0)^\triangleright \coprod_{\Delta^0} S \to S?$And does this natural map exist whenever, for example in the case $X^\triangleright \coprod_X S$, $X$ has a terminal object (or strongly terminal object)?

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