  Let $\phi: G \rightarrow H$ be a group homomorphism such that $H$ is a
totally disconnected locally compact (t.d.l.c.) group and the image of $\phi$
is dense. We show that all such homomorphisms arise as completions of $G$ with
respect to uniformities of a particular kind. Moreover, $H$ is determined up to
a compact normal subgroup by the pair $(G,\phi^{-1}(L))$, where $L$ is a
compact open subgroup of $H$. These results generalize the well-known
properties of profinite completions to the locally compact setting.
