To lay down, to deposit, to place — and by extension, to grant a favor or earn goodwill. Used in Acts of political leaders 'depositing favors' with the Jewish leadership.
In Acts 24:27, Felix left Paul in prison to 'grant a favor' (katatithēmi) to the Jews. This word exposes the corruption of justice systems that trade human lives for political capital. Paul — innocent — remains imprisoned because a governor wanted to make a deposit in his political favor bank. The word indicts every system that treats persons as bargaining chips and reminds us that true justice is impartial (Deut 1:17).