Profinite completion ($\hat{G}$)

The profinite completion of a group $G$ is the inverse limit of its finite quotient groups:The canonical map $G\to \hat{G}$ has dense image and is injective exactly when $G$ is residually finite.

Every homomorphism from a group $G$ to a profinite group $P$ extends uniquely to a continuous homomorphism $\hat{G}\to P$. 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 $\hat{G}$.