Usage: 2 forms in 2 verses
Form Translated As     In Verse(s)
καθωμολογήσατο solemnly promise LXX.Exod.21.8
καθομολογήσηται he solemnly promised LXX.Exod.21.9

Liddell-Scott-Jones

καθομολογέω (kathomologeō) to promise G:V

καθομολογέω-έω,

confess, allow, especially to one's detriment, [Refs 5th c.BC+]

__2 consent, absolutely, [Refs 5th c.BC+]
__2.b consent to accept in payment, τόκους [Refs 4th c.BC+]
__II promise, engage, vow:—passive, ἀνάθημα τῷ θεῷ [Refs 2nd c.AD+]
__II.2 betroth, τινὰ τῷ υἱῷ [LXX]; Κλαυδίαν ἀνδρί [Refs 1st c.AD+]:—perfect passive in middle sense, τὴν ἀδελφὴν.. γυναῖκά τινι καθωμολογημένος [Refs]; but in passive sense, [Refs]: pluperfect καθωμολόγητο [Refs 1st c.BC+]
__II.3 pledge, mortgage, ὑποθήκην κ. τί τινι [Refs 4th c.AD+]
__III middle in sense of active [Refs]future infinitive, [Refs 1st c.BC+]

LSJ Scripture Refs: LXX.Exod.21.9