Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 5 / b

For every open normal subgroup $U◃G$, the compositehas finite image and therefore factors uniquely through the profinite completion $\hat{\Gamma }$. These factor maps are compatible as $U$ varies. The universal property of an inverse limit consequently produces a continuous homomorphismwith $\hat{f}\iota =f$. It is unique because $\iota (\Gamma )$ is dense in $\hat{\Gamma }$ and two continuous maps into the Hausdorff group $G$ that agree on a dense subset agree everywhere. This is the universal property of profinite completion.