gift brought by a visitor
Honorific abbreviation of お持たせ物. Used when offering a guest something from the gift they brought, e.g., 'Please have some of what you brought.'
See also: お持たせ物
I'm sorry to offer you something you brought yourself, but please have it with your tea.
Shall we open the gift you brought?
gift received from a visitor
Abbreviation of お持たせ物 (o-motase-mono), from the honorific prefix お + 持たせ (causative form of 持つ 'to hold/carry') + 物 'thing', literally 'something made to be brought'.