In general, if $A\subset B$ is a full reflective subcategory, then each object $a\in A$ is isomorphic to its image under the reflector. This seems to include many cases: $\mathbf{Ab}\subset \mathbf{Grp}$, $\mathbf{Met}\subset\mathbf{CompMet}$, \mathbf{CompMet}\subset\mathbf{Met}$, $\mathbf{Top}_{n+1}\subseteq \mathbf{Top}_{n}$ (as in http://mathoverflow.net/questions/9504/why-is-top4-a-reflective-subcategory-of-top3), etc.
EDIT: Following Pete L. Clark's comment, here is a clarification: The subcategory $A$ above is called reflective if the inclusion functor $A\subset B$ has a left adjoint, and full if this inclusion functor is full. In case $A$ is a reflective subcategory, the left adjoint to the inclusion functor is called a reflector.
