What should the definition of "Yoneda property" be? 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.
 A: Hi Jim, not sure if this is the sort of thing you are after, but here is one possibility. 
Let $K$ be a category and $F=(f_i:A_i\to B_i)_{i\in I}$ a family of morphisms in $K$. Say that an object $X$ is injective to $F$ if for each $i\in I$ and each $a:A_i\to X$ there exists a morphism $b:B_i\to X$ whose restriction along $f_i$ is $a$. The collection of all such objects $X$ (for given $F$) is called an injectivity class in $K$. 
Any injectivity class in $[C^{op},Set]$ defines a property of objects of $C$: those objects $c$ for which the representable functor $C(-,c)$ lies in the injectivity class. Your various examples arise in this way:


*

*For injectivity, just take all maps of the form $C(-,i):C(-,a)\to C(-,b)$ with $i:a\to b$ mono. 

*For terminal object, take all the maps $0\to C(-,a)$ and $\nabla:C(-,a+a)\to C(-,a)$ (where $\nabla$ is the codiagonal)

*and similarly being formally smooth or formally unramified. 


Your properties "of uniqueness type'' can be seen as a special case of those "of existence type'', by using codiagonal maps, as in the case of terminal object. 
Of course in the first example, of injectivity, you could just as well work in the original category $C$ itself, rather than $[C^{op},Set]$. More generally, you would always be able to do this if $C$ had colimits. 
You will get better properties if the family of morphisms defining the injectivity class is, or can be taken to be, small. Not sure if this is important for your purposes. 
A: 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, $X\times Y$ is a product if for any $Z\to X, Z\to Y$ we get $Z\to X\times Y$ 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.
