Universal property of profinite completion
= Universal property of profinite completion
Every homomorphism from a group $G$ to a <profinite group> $P$ extends uniquely to a continuous homomorphism $\widehat G\to P$. Construct the extension on every finite quotient of $P$ and invoke the <universal property of an inverse limit>; uniqueness follows from the density of $G$ in $\widehat G$.