Proposition 2

(a) If $f : X \to X'$ and $g : X' \to X''$ are two uniformly continuous mappings, then $g \circ f : X \to X''$ is uniformly continuous.

(b) A bijection $f$ of a uniform space X onto a uniform space X' is an isomorphism if and only if $f$ and the inverse of $f$ are uniformly continuous.

This follows immediately from the interpretation of Definition 1 in terms of the product mapping $f \times f$.