Operating and cooperating grace: Augustine's words
Thesis XLII. In the Roman schools, too, the division of grace into operating and cooperating is familiar and very well known. It is drawn from Augustine's words in On Grace and Free Choice, chapter 17: 'He works at the start, so that we may will; and he cooperates with those who will, to bring the work to completion.' A little later he says: 'He works without us so that we may will; but when we will, and will in such a way that we act, he cooperates with us.'
So on a common and rather loose understanding, operating grace is the grace that works alone, without us, and cooperating grace is the grace that does not work without us but with which we cooperate.
Latin
XLII. In Scholis quoque Romanis trita & celeberrima est gratiae divisio in Operantem, & Cooperantem. Quae sumpta est ex verbis Augustini, in libro de Gratia & libero Arbitrio capite decimo septimo. Ipse, inquit, ut velimus operatur incipiens, qui volentibus cooperatur perficiens. Et paulo post, ut velimus, inquit, sine nobis operatur. Quum autem volumus, & sic volumus, ut faciamus, nobiscum cooperatur. Adeoque juxta communem & confusam quandam notionem, Gratia Operans dicitur, quae sola sine nobis operatur. Cooperans vero, quae non operatur sine nobis, sed cui cooperamur.