By Corollary 3.2 it suffices to show that $\lim_{\mathcal{X}}\ev\circ p_{\mathcal{X}}=\ev$, which amounts to proving $\lim_{\mathcal{X}}\ev_{\mathcal{Y}}\circ p_{\mathcal{X}}=\ev_{\mathcal{Y}}$ for each $\mathcal{Y}$. This follows from (i) of Proposition 3.5. ∎