**Affine schemes**:

Given any ring $R$, try to find a map from it into local ring $L$ which is initial among maps to local rings (i.e. any other map from $R$ into a local ring should factor through this one, followed by a *map of local rings*, i.e. one such that the preimage of the maximal ideal is the maximal ideal). Such a thing does not exist, unless $R$ is already local.

But if we allow $L$ to be a ring object living in a different topos than that of sets, then it exists: It is the local ring object living in $Sh(Spec R)$ given by the structure sheaf $\mathcal{O}_{Spec R}$ (see also my post [here][1])


  [1]: https://mathoverflow.net/questions/8204/how-can-i-really-motivate-the-zariski-topology-on-a-scheme/14334#14334