So Yoneda says objects are fully determined by maps from all objects. However, sometimes we can do better. For instance, Yoneda also says that presheaves are fully determined by maps from representables and schemes are fully determined by maps from affine schemes and so on (1/?)

Anyway, is there a name for the kind of subcategory that does this? Also, notably both examples (and every other one I can think of) have the objects as glued together versions of the special objects. Is that true in general? I.e. if C’ is a subcategory of C (2/?)

such that the inclusion followed by the Yoneda embedding is conservative (or just isomorphism creating), is every object of C a colimit of a diagram in C’? Disclaimer, I don’t even fully know if that’s true for schemes; I have no clue how tf algebraic geometry works (3/?, ?=3)

Correction: I don’t mean inclusion followed by Yoneda embedding I mean Y.E. followed by the precomoposition of the inclusion.