Let $F$ be a free group with $r$ generators. Then for every $n \in \mathbb{N}$ there are exactly $H_n = \left(\frac{n!}{2}\right)^r - 1$ nontrivial distinct homomorphisms from $F$ to $A_n$, the alternating group on $n$ elements. There is a canonical map $$\iota \colon F \to \mathcal{A} = \prod_{n = 1}^\infty A_n^{H_n},$$ where $A_n^{H_n}$ denotes the $H_n$-fold cartesian product of $A_n$. Following some classical results one can verify that this is actually an embedding. Clearly, $\mathcal{A}$ is a profinite group so it makes sense to consider the closure of $\iota(F)$ in $\mathcal{A}$. In a paper by Glebsky (see http://arxiv.org/abs/1506.06940) it is briefly mentioned that this completion, let's denote it $\hat{F}_\mathcal{A}$ - the pro-alternating completion of $F$, is actually isomorphic to $\hat{F}$, the profinite completion of $F$. I fail to see why this is the case.
Would anyone mind shedding some light into this either by giving a brief explanation of by giving a reference to some readable material?