The Stacks project

I understand why $U_{X/S}$ is cocontinuous, but why is it continuous?

For example, suppose $S = \text{Spec} (A, I, \gamma) =\text{Spec } (\mathbb Z/p^2, 0, 0)$ and $X = \text{Spec } A$. Take $U_1 = T_1 = \text{Spec } \mathbb Z/p^2$ with the trivial PD structure, and $T_2 = (\text{Spec } \mathbb Z/p^2, p, \delta)$ with the standard PD structure on $p$, so $U_2 \hookrightarrow T_2$ is the closed point. Then $U_{X/S}$ should send the cover $\{ T_2 \rightarrow T_1 \}$ to $\{U_2 \rightarrow U_1\}$, which doesn't seem like a cover to me.

Did I miss something?

Yes, this is a mistake. Thanks very much for pointing this out. I have fixed this here.