Corollary

Let $X$ be a set, let $(Y_i)_{i \in I}$ be a family of uniform spaces and let $\mathscr{S}$ be a set of subsets of $X$. If we endow $\prod_{i \in I} Y_i$ with the product uniformity, the canonical bijection of the uniform space $\mathscr{F}_\mathscr{S}(X, \prod_{i \in I} Y_i)$ onto the product uniform space $\prod_{i \in I} \mathscr{F}_\mathscr{S}(X; Y_i)$ (Set Theory, R, § 4, no. 13) is an isomorphism.