An endomorphism of a finite set (or finite-dimensional vector space) is an injection if and only if it is a surjection.

For sets and vector spaces, injections are the same as monomorphisms, and surjections are the same as epimorphisms. So these are very naturally representable/co-representable properties. (A map $X\to Y$ is said to be a monomorphism if the maps $\mathrm{Hom}(Z,X)\to\mathrm{Hom}(Z,Y)$ are injective for all $Z$, and it's an epimorphism if the maps $\mathrm{Hom}(X,Z)\to\mathrm{Hom}(Y,Z)$ are all injections.)

NB This isn't technically an answer to the question, since these are properties of morphisms and not of objects, but it has the right spirit.