Exercise 9, § 2

Let $X$ be a set, $Y$ a uniform space and $S$ a covering of $X$. A subset $H$ of the uniform space $\mathscr{J}_S(X; Y)$ is precompact if and only if (i) for each $x \in X$, $H(x)$ is precompact in $Y$; (ii) the uniformity of $S$-convergence and that of pointwise convergence coincide on $H$. (Apply Theorem 2 by considering $X$ as a discrete space and reducing to the case where $Y$ is Hausdorff and complete.)