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Firstly:It is not prescribed to offer or give the udhiyah on behalf of the Prophet (blessings and peace of Allah be upon him), because that was not narrated from any of the Sahaabah, despite their perfect love for him and their perfect eagerness to do good, and the Prophet (blessings and peace of Allah be upon him) did not instruct his ummah to do that, as he instructed them to send blessings upon him and to pray for al-waseelah (a station in Paradise) and al-fadeelah (a rank above the rest of creation) for him after the adhaan. If it were something good, he (blessings and peace of Allah be upon him) would have directed them to do it. Moreover, whatever good his ummah does, he is rewarded for it, because he was the one who guided, taught and called people to it. So for the doer to dedicate the reward of the deed to the Prophet (blessings and peace of Allah be upon him) is pointless; rather it means that the doer is deflecting the reward from himself without that benefiting anyone else.
This has been explained in detail in the answer to question no. 52772
Secondly:
Offering udhiyah on behalf of the dead may occur in one of three ways:
1.
Where the udhiyah is offered on their behalf along with the living, such as if a man offers an udhiyah on behalf of himself and on behalf of the members of his family, intending it to include both the living and the dead. This is permissible, and the evidence for that is the fact that the Prophet (blessings and peace of Allah be upon him) used to offer the udhiyah on behalf of himself and on behalf of the members of his family, including those who had previously died, such as Khadeejah (may Allah be pleased with her).
2.
Where the udhiyah is offered on behalf of the dead in fulfilment of their wishes and instructions given before their death (bequests). This is obligatory, except in cases where one is not able to do that. The evidence for that is the verse in which Allah, may He be exalted, says concerning the changing of bequests (interpretation of the meaning):
“Then whoever changes the bequest after hearing it, the sin shall be on those who make the change. Truly, Allah is All-Hearer, All-Knower”
[al-Baqarah 2:181].
3.
Where the udhiyah is offered on behalf of the dead voluntarily and independently of the living (such as when a person offers a separate udhiyah on behalf of his father, or a separate udhiyah on behalf of his mother). This is permissible, and the Hanbali fuqaha’ stated that the reward thereof will reach the deceased and benefit him, by analogy with giving charity on his behalf. But it is preferable not to do that, because there is no report to that effect. The Prophet (blessings and peace of Allah be upon him) did not offer an udhiyah specifically on behalf of any of his deceased loved ones. He did not offer an udhiyah on behalf of his paternal uncle Hamzah, who was one of the dearest of his relatives to him, or on behalf of his children who died during his lifetime, or on behalf of his wife Khadeejah, who was one of the dearest of his wives to him. And there is no report from any of his companions during his lifetime that any of them offered an udhiyah on behalf of one of his deceased loved ones.
See also the answer to question no. 36596.
And Allah knows best.