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 <a href="http://mishap.sdf.org/by:gavrilovich/mints-lifting-property-as-negation-DMG_5_no_4_dec_2014.pdf">an unfinished update.</a>