Homomorphisms into totally disconnected, locally compact groups with dense image

Let phi : G -> 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.