ct.category theory - What should the definition of "Yoneda property" be?

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 $forall X$, $F(X)$ has property $P'$

as you'd suggested above. I think we can avoid $P'$ being a natural isomorphism with $h_X$ 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, $Xtimes Y$ is a product if for any $Zto X, Zto Y$ we get $Zto Xtimes Y$ 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 $F(X)=G(X)times H(X)$" 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.