$\def\codom{\operatorname{codom}}$If $G$ is a first-countable abelian topological group, one can find a morphism $G\to\hat{G}$ of abelian topological groups, with $\hat{G}$ first-countable and sequentially complete (by definition, Hausdorff and such that every Cauchy sequence converges) that is universal among morphisms of $G$ into sequentially complete topological abelian groups. Both $\hat{G}$ and the morphism $G\to\hat{G}$ have a very concrete description. Namely, $\hat{G}$ is the set of Cauchy sequences in $G$ modulo the sequences converging to zero, and $\phi:G\to\hat{G}$ sends $g\in G$ to the class of the sequence constantly equal to $g$. See Lemma 9 here. Also, $\phi(G)$ is dense in $\hat{G}$ and $\ker\phi=\overline{\{0\}}\subset G$.
I wonder: can we characterize this universality? By this I mean:
Suppose $\rho:G\to H$ is a morphism of topological abelian groups such that $H$ is first-countable sequentially complete, $\ker\rho=\overline{\{0\}}\subset G$ and $\rho(G)$ is dense in $H$. Is then $\rho$ universal among morphisms of $G$ into sequentially complete topological abelian groups?
I tried two approaches: (a) redoing the proof of Lemma 9 from last link but now for $\rho$ instead for $\phi$ and (b) considering the unique factorization $\tilde{\rho}:\hat{G}\to H$ and trying to prove that it is an isomorphism. I was always unsuccessful: with (a), I considered $\psi:G\to K$, where $K$ is sequentially complete. Then to define a factorization $\overline{\psi}:H\to K$ through $\rho$, for $h\in H$, one picks $x_n\in G$ with $\rho(x_n)\to h $ and then says $\overline{\psi}(h)=\lim \psi(x_n)$. The problem is: how do we know that $(\psi(x_n))$ is convergent in $K$? It would suffice to show $(x_n)$ is Cauchy in $G$ (or that we can pick such $(x_n)$ Cauchy), by why is this the case? For (b), I don't know how to show surjectiveness of $\tilde{\rho}$ because of the same problem.
If we further assume that every element of $H$ is the limit of the image of a Cauchy sequence in $G$, one could now show well-definedness in (a) and surjectiveness in (b). But now I'm helpless showing continuity in (a) and injectivity and openness in (b).
Am I missing some property that we should ask $\rho$ to have?
