Definition 1

A mapping $f$ of a uniform space $X$ into a uniform space $X'$ is said to be uniformly continuous if, for each entourage $V'$ of $X'$, there is an entourage $V$ of $X$ such that the relation $(x, y) \in V$ implies $(f(x), f(y)) \in V'$.

In more expressive terms we may say that $f$ is uniformly continuous if $f(x)$ and $f(y)$ are as close to each other as we please whenever $x$ and $y$ are close enough.

If we put $g = f \times f$, then Definition 1 means that whenever $V'$ is an entourage of $X'$, $\overline{g^{-1}}(V')$ is an entourage of $X$.