Let $C$ be a category. I'd like to say that a property $P$ of objects of $C$ (or rather isomorphism classes of objects) is a "Yoneda property" or a "maps-in property" if there is a property $P'$ of contravariant functors $h:C\to\mathrm{Set}$ such that the functor $\mathrm{Hom}(-,X)$ has $P'$ if and only if $X$ has $P$. We might also say a property is "co-Yoneda" or "maps-out" if the induced property on $C^{\mathrm{op}}$ is Yoneda.

But this definition is useless because every property is a Yoneda property (and, hence, also a co-Yoneda property) -- just take $P'$ to be the property "$h$ is isomorphic to the functor $\mathrm{Hom}(-,X)$, for some object $X$ with property $P$". So my question is: Is there a good definition of "Yoneda property"?

Here are some examples of what I have in mind. In the category of modules over a given ring, injectivity should be a Yoneda property and projectivity should be a co-Yoneda property. In any category, being a terminal object should be a Yoneda property and being an initial object should be a co-Yoneda property. We could do the same thing with maps instead of objects, and then in the category of schemes being proper should be a Yoneda property (by the valuative criterion), as would being separated, formally smooth, formally unramified, locally of finite presentation, and so on.

A few more remarks:

It seems that we'd want to keep the definition I gave above but make some restriction on properties of functors we allow. Quantifying existentially over all objects of the category (which is what breaks the definition above) probably should not be allowed. But what exactly should be allowed?

There appear to be different kinds of Yoneda properties. For example, in the category of schemes, the definition of formally smooth is of the form "for all diagrams of type $Y$, there exists a map $f$ such that $Z$ holds", and the definition of formally unramified is of the form "for all diagrams of type $Y$ and all maps $f, f'$ such that $Z$ holds, we have $f=f'$". Maybe it would be better to distinguish these different kinds of properties. So it might be more natural to define separately "Yoneda properties of existence type" (e.g. formal smoothness), "Yoneda properties of uniqueness type" (formal unramifiedness), and maybe others.