Is there a universal definition of what that phrase should mean?
Suppose we are considering objects of a topos with a particular property, call them $P$-objects (I have in mind the case where P="compact", as in Johnstone's Topos Theory). What does the phrase "the topos is generated by its $P$-objects" mean? It looks like it usually means that every object is a coproduct of two $P$-objects. But I think sometimes it means something like "every object is a filtered/cofiltered limit/colimit of $P$-objects".