an $n$-simplex of $\mathcal{C}_{/x}$ is given by an $(n+1)$-simplex of $\mathcal{C}$ with final vertex $x$, obtain from this an $(n+2)$-simplex of $\mathcal{C}$ by degenerating the edge from $n+1$ to $n+2$, curry this to an $(n+1)$-simplex of $\mathcal{C}_{/x}$ and apply the functor to obtain an $(n+1)$-simplex of $\mathcal{C}_{/y}$, which curries to an $(n+2)$-simplex of $\mathcal{C}$ whose edge from $n+1$ to $n+2$ is given by a morphism $f\colon x\rightarrow y$ (the image of $\mathrm{id}_x$ under the functor), so this curries to an $n$-simplex of $\mathcal{C}_{/f}$.