Every continuous map from a nonempty compact Hausdorff space to a uniform space is uniformly continuous

Let $V$ be a target entourage and choose a symmetric $W$ with $W^{-1}\circ W=W^{\circ2}\subseteq V$. For each $x\in X$, let $O_x$ be the union of all open sets $O$ such that $x\in O$ and $f[O]\subseteq W[f(x)]$. Continuity makes this family nonempty, and its union is an open neighbourhood of $x$ satisfying $f[O_x]\subseteq W[f(x)]$.

The open cover $(O_x)_{x\in X}$ is uniform by [L3]. Hence there is a source entourage $E$ whose ball cover refines it: for each $a\in X$, some $O_x$ contains $E[a]$.

If $(a,b)\in E$, then $a,b\in E[a]\subseteq O_x$ for some $x$. Thus $f(a),f(b)\in W[f(x)]$, so $(f(a),f(b))\in W^{-1}\circ W\subseteq V$. This is uniform continuity.