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It seems, some obscure terminology.
Yeah, it’s not particularly convincing terminology. Also, it clashes with the terminology of Z-categories.
Moreover, one wonders whether small (pre)sheaves, rather than all presheaves, might give the better concept, without running into size issues.
I had a vague memory of this and I checked now, the terminology originates from Demazure-Gabriel famous functorial treatise on algebraic geometry and algebraic groups. They work with universes so the smallness is accounted for appropriately.
I meant small in the sense of being a small colimit of representables, not valued in small sets, in case that wasn’t clear. I see from this MathOverflow answer D&G are imposing the constraint that a Z-functor is only defined on a subcategory of “very small” rings, relative to the smaller of a pair of universes.
6 Oh, right, I forgot about that.
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