which means: to take responsibility for a situation while another person is temporarily absent. To “hold someone down” in a relationship loosely means to be willing to take care of this person financially, mentally, and/or spiritually when that person is unable to do so for themselves.
If someone makes a statement of intent and you respond by saying 'I will hold you to that', you are either genuinely expecting them to keep that promise or else you are saying it ironically because you don't really expect them to follow through.