Let $S$ be a set, let $(Y,e)$ and $(Z,d)$ be metric spaces, and let $\Phi:Y\to Z$ be uniformly continuous. Let $(f_k)_{k\in\mathbb N}$ be a sequence of maps $f_k:S\to Y$, and let $f:S\to Y$ be a map. If $f_k$ converges uniformly to $f$ on $S$ with respect to the metric $e$, then $\Phi\circ f_k:S\to Z$ converges uniformly to $\Phi\circ f:S\to Z$ with respect to the metric $d$.