Well, I'll take a stab at it. I think you're going to want properties P of functors F of the form
F has property P if and only if forallX, F(X) has property P′
as you'd suggested above. I think we can avoid P′ being a natural isomorphism with hX by making sure it's a property of all of these sets, rather than something we say about it as a functor (on the other hand, if we want to define Yoneda and CoYoneda properties of morphisms, this becomes a problem, and so this is really bothering me philosophically...)
But this definition handles "is a group", "is initial", "is terminal" by "every F(X) is a group", and statements that for a contravariant or covariant (as appropriate) we have a single element. We're going to want to allow products as well to be Yoneda, presumably. Now, XtimesY is a product if for any ZtoX,ZtoY we get ZtoXtimesY commuting with projection. So I guess here we might want to have the property "There exist functors G,H such that for all X, we have F(X)=G(X)timesH(X)" perhaps? It avoids quantifying over the category in question, but instead does so over the category of functors, which I haven't thought through terribly well.
Mostly I'm just thinking aloud here, but no one else had much, so I thought I'd throw my two cents in.
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