Remark 1

The general definition of the entourages of an initial uniformity (Chapter II, § 2, no. 3, Proposition 4) shows that a fundamental system of entourages of $\mathscr{F}_\mathscr{S}(X; Y)$ may be obtained as follows: for each $A \in \mathscr{S}$ and each entourage $V$ of a fundamental system of entourages $\mathscr{B}$ of $Y$, let $W(A, V)$ be the set of all pairs of mappings $(u, v)$ of $X$ into $Y$ such that $(u(x), v(x)) \in V$ for each $x \in A$; as $A$ runs through $\mathscr{S}$ and $V$ runs through $\mathscr{B}$, the finite intersections of the sets $W(A, V)$ form a fundamental system of entourages of $\mathscr{F}_\mathscr{S}(X; Y)$.

This description shows immediately that if $\mathscr{S}, \mathscr{S}'$ are two sets of subsets of $X$ such that $\mathscr{S} \subset \mathscr{S}'$, then the uniformity of $\mathscr{S}'$-convergence is finer than that of $\mathscr{S}$-convergence.