Proposition 3

Let $X$ be a topological (resp. uniform) space, let $(Y_i)_{i \in I}$ be a family of uniform spaces, let $Y$ be a set, and for each $i \in I$, let $f_i$ be a mapping of $Y$ into $Y_i$. Let $Y$ be endowed with the coarsest uniformity for which all the $f_i$ are uniformly continuous. For a subset $H$ of $\mathscr{F}(X; Y)$ to be equicontinuous (resp. uniformly equicontinuous) it is necessary and sufficient that, for each $i \in I$, the image of $H$ under the mapping $u \to f_i \circ u$ be an equicontinuous (resp. uniformly equicontinuous) subset of $\mathscr{F}(X; Y_i)$.

This is an immediate consequence of Definitions 1 and 2 and the definition of the entourages of $Y$.