Profinite Completion of Free Pro-$\mathcal{C}$ Groups

We investigate the ability of a free pro-$\text{CC}$ group of infinite rank to abstractly solve abstract embedding problems, and conclude that for some varieties $\text{CC}$, the profinite completion of any order, of a free pro-$\text{CC}$ group of infinite rank, is a free pro-$\text{CC}$ group as well, of the corresponding rank.