$\pi_k: \prod G_i \to G_k$ defined by $\pi_k((g_i))=g_k$. Usually, to prove some $f$'s well definedness, show $x=y \implies f(x)=f(y).$ There's an obvious difficulty here, there can be many different sequence with the same $k$'th coordinate.
(The map is an epimorphism of groups.)