\mathcal{A} possesses a generator, i.e. there is an object G in \mathcal{A} such that \operatorname{Hom}(G,-) is a faithful functor from \mathcal{A} to the category of sets.
The name "Grothendieck category" neither appeared in Grothendieck's Tôhoku paper[REF] nor in Gabriel's thesis;[REF] it came into use in the second half of the 1960s by authors including J.-E.