Imam Tirmidhi's Ism (proper name) was Muhammad. His Nasab (full name including his genealogical chain) was: Muhammad b. 'Isa b. Thawra b. Musa b. al-Dahhak. Imam Tirmidhi's Kunya (honorific name) was Abu 'Isa. Certain scholars raised objections regarding his kunya, since there is a tradition in the Musannaf of ibn Abu Shaiba, (a particular type of Hadith book) where Prophet Muhammad (SAW) reprimanded someone whose name was Abu 'Isa by saying that Isa had no father, implying that it was not becoming to keep the kunya Abu 'Isa. Hence the question arises regarding Tirmidhi's doing so. The permissibility for this practice is established in Sunan Abu Dawud, where it is narrated that Muqhira b. Sh'uba (d.50 A.H.) who was a companion of the Prophet (SAW), had the kunya Abu 'Isa. Once, 'Umar b. Khattab (d.24 A.H.) the second Caliph of Oslam rebuked him by saying, "Does it not suffice you to be called Abu 'Abd AllAh". Mughira replied, "It was the Prophet (SAW) of Allah who gave me this kunya".