Jamieson, Fausset and Brown on Hebrews 7:22

22. surety--ensuring in His own person the certainty of the covenant to us. This He did by becoming responsible for our guilt, by sealing the covenant with His blood, and by being openly acknowledged as our triumphant Saviour by the Father, who raised Him from the dead. Thus He is at once God's surety for man, and man's surety for God, and so Mediator between God and man (He 8:6).

better--He 8:6; 13:20, "everlasting."

testament--sometimes translated, "covenant." The Greek term implies that it is appointed by God, and comprises the relations and bearings partly of a covenant, partly of a testament: (1) the appointment made without the concurrence of a second party, of somewhat concerning that second party; a last will or testament, so in He 9:16, 17; (2) a mutual agreement in which both parties consent.