Proposition 3

a) Let $X$ be a set; let $\mathscr{S}$ be a set of subsets of $X$; let $Y, Y'$ be two uniform spaces; and let $f : Y \to Y'$ be a uniformly continuous mapping. Then the mapping $u \to f \circ u$ of $\mathscr{F}_\mathscr{S}(X; Y)$ into $\mathscr{F}_\mathscr{S}(X; Y')$ is uniformly continuous.

b) Let $X, X'$ be two sets; let $\mathscr{S}$ (resp. $\mathscr{S}'$) be a set of subsets of $X$ (resp. $X'$); let $Y$ be a uniform space; and let $g : X' \to X$ be a mapping such that, for each $A' \in \mathscr{S}'$, $g(A')$ is contained in a finite union of sets of $\mathscr{S}$. Then the mapping $u \to u \circ g$ of $\mathscr{F}_\mathscr{S}(X, Y)$ into $\mathscr{F}_{\mathscr{S}'}(X'; Y)$ is uniformly continuous.