My questions refer to the following article (both refer to page 27):
https://arxiv.org/pdf/2002.10278.pdf
In the article we find the statement that for a non-abelian limit group $L$ we always find a sequence of epimorphisms to $F_2$ that converges to $L$. By definition of a (non-abelian) limit group we find -a priori- only a sequence of homomorphisms. Why can the homomorphisms (in this situation) always be chosen to be surjective? From the article "An Introduction to Limit Groups" (Wilton) I could find out that limit groups are fully residually free and because $L$ is not abelian it has to contain a free group of rank at least 2. So there exists an epimorphism from $L$ to $F_2$. But this does not answer why this convergent homomorphism can be chosen to be epimorphisms.
On page 27 of the article we have a sequence $g_n$ of epimorphisms from $F_l$ to $L$. The autors say that we can pass to a subsequence that converges into a Limit group $L_1$. Why can one choose such a subsequence? I think that the answer to that is not so difficult but I still do not have a real idea. Maybe one can again use that limit groups are fully residually free and can apply properties of fully residually free groups to get the statement.