The name of T Smith is recorded on the Stockport War Memorial amongst those serving with the Royal Engineers. However, it is possible that there is an error and the man intended to be commemorated was a Joseph Smith. However, with the passage of time, it is impossible to be absolutely certain.
In its edition of 19 April 1918, the Stockport Advertiser reported "Sapper T Smith, Royal Engineers, aged 45, died in hospital in France on April 8 of wounds received in action. He was married and lived at 38 Robinson Street, Stockport." Other than this newspaper report, it has not been possible to find any mention of a T Smith who died on this date in the records of the Commonwelath War Graves Commission or the CD-ROM "Soldiers Died in the Great War".
However, the research revealed the death of a Sapper Joseph Smith on this date. His local connection was clear - military records indicate he had been born in Hazel Grove and had enlisted into the Army at Stockport. Joseph Smith is buried at Bois Guillaume Cemetery which is a suburb of the French town of Rouen. Burials in this cemetery were made by the Army's adjacent No. 8 General Hospital. Joseph Smith had died of wounds received whilst serving with 258th Tunnelling Company, Royal Engineers
With such a common name as Smith, it is difficult to be sure of a correct identification in records, but the matter is further complicated by information from the family history website, CheshireBMD. This records the birth in Hazel Grove of a Thomas Smith in 1873, but has no record of a Joseph Smith of the right age. It is possible that this is the same man whose correct first name was Thomas but who chose, for some reason, to enlist under the false name of Joseph. It was not unheard of.