Imam Ibn al-Kathir (may Allah have mercy upon him) wrote:

“The Mention of the Death of Adam and His Legacy to his Son Sheeth (may peace be upon them both)

And the meaning of Sheeth is the gift of Allah. He was named that because they (Adam and Hawwa) were granted him after the death of Haabil. When Adam died (may peace be upon him) it was on Friday. The Angels came with Hanut and shrouds from Allah, the Exalted from Jannah. They ascribed his son to him and he gave the legacy to Sheeth (may peace be upon him)” [Saheeh Qassas al-Anbiya pg. 43]

Then He mentions a narration from Ubayy bin Ka’ab (may Allah be pleased with him) about the death of Adam and his sons. The narration is reported by Abdullah bin Ahmad in Zawaaid al-Musnad (5/136) and others. Hafidh Ibn Kathir graded it Saheeh himself and Shaikh Saleem al-Hilali graded it Saheeh in his book Saheeh Qassas al-Anbiya . He said it is also reported in Marfoo’ form and the narration in Mawqoof form takes the ruling of Marfoo’. [See: Saheeh Qassas al-Anbiya of Shaikh Saleem (pg. 43)]

In terms of Sheeth being a Prophet, then Hafidh Ibn Kathir wrote:

“When Adam passed (may peace be upon him) passed away, Sheeth established the responsibility of the affair. He was a Prophet according to the text of the Hadith which is reported by Ibn Hibban in his Saheeh on the authority of Abu Dharr in marfoo’ form that Fifty scrolls were revealed to him (i.e. Sheeth)” [Bidaaya wan-Nihaayah (1/230)]

However, this Hadith cited by Ibn Kathir was graded very weak by Imam al-Albani in Ta’liqaat al-Hisaan ‘ala Saheeh Ibn Hibban (1/387).

So while we know he is a son of Adam, we do not have authentic proof to say he was a Prophet of Allah.

And Allah Knows Best

Faisal bin Abdul Qaadir bin Hassan
Abu Sulaymaan