There are a few examples based on the lifting property in the category of topological spaces. For example, see answers there http://math.stackexchange.com/questions/434312/are-there-useful-categorical-characterisations-of-the-topological-separation-axioms

But there are also other examples, see http://arxiv.org/pdf/1408.6710.pdf or more examples in an unfinished update.

There are a few examples based on the lifting property in the category of topological spaces. For example, see answers there https://math.stackexchange.com/questions/434312/are-there-useful-categorical-characterisations-of-the-topological-separation-axioms

But there are also other examples, see http://arxiv.org/pdf/1408.6710.pdf or more examples in an unfinished update.