Is it permissible to offer an Uḍḥiyah (sacrificial animal) on behalf of the deceased or on behalf of the Prophet (peace be upon him)?

It is not prescribed to offer a sacrificial animal (udhiyah) as a gift to the Prophet (peace be upon him) or on his behalf, because this was not transmitted from the Companions. Had it been good, he would have guided them to it. Every good deed performed by the Ummah (Muslim community) reaches the Prophet (peace be upon him).

As for offering a sacrificial animal on behalf of the deceased, there are three scenarios: 1. Offering it on their behalf as a subsidiary to the living: This is when a man offers the sacrifice on behalf of himself and his household, intending both the living and the deceased. This is permissible. 2. Offering it on behalf of the deceased in accordance with their wills: This is obligatory, as it is the fulfillment of a will. 3. Offering it on behalf of the deceased as an independent voluntary act: This is permissible, and its reward reaches the deceased. However, it is preferable to avoid it because it has not been transmitted (from the early generations).