I like this discussion by Mike Shulman. And a propos of the related discussion of terminology I note the terms here describe mathematical features (very well, I think) rather than focusing on whether one *likes* the features. 2010/10/2 Michael Shulman <shulman@math.uchicago.edu>:
I personally prefer to say that "unique choice structure" is something "in between" property and structure. Kelly and Lack dubbed it "Property-like structure" in their paper with that title. The difference is exactly as you say: property-like structure is unique (up to unique isomorphism) when it exists, but is not necessarily "preserved" by all morphisms. In terms of forgetful functors, property-like structure corresponds to a functor which is *pseudomonic*, i.e. faithful, and full-on-isomorphisms. Another nice example is that being a monoid is a "property" of a semigroup, i.e. a semigroup can have at most one identity element, but a semigroup homomorphism between monoids need not be a monoid homomorphism.
best, Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]