Remark 2

If $f : X \to X'$ is a uniformly continuous mapping, then $f$ remains uniformly continuous if we replace the uniformity of $X$ by a finer uniformity and that of $X'$ by a coarser uniformity (no. 1, Proposition 2). In other words, the finer the uniformity of $X$ and the coarser the uniformity of $X'$, the more uniformly continuous mappings there are of $X$ into $X'$.