Let $\mathit{Profinite}_{\mathrm{Ab}}$ be the category of profinite abelian groups, and let $\mathit{Profinite}_{\mathrm{Set}}$ be the category of profinite sets. Does the forgetful functor

$$\mathit{Profinite}_{\mathrm{Ab}} \to \mathit{Profinite}_{Sets}$$

admit a left adjoint?

I am a beginner to this kind of question; I do not even know if both the domain and codomain categories admit all small colimits.