Officials released Jaffa's name on Wednesday and confirmed that he died from exposure.
Jaffa, 55, was known to hang around the area of the Eaton Centre.
His death from the temperatures was the second in less than 24 hours, prompting a protest at mayor John Tory's office and a city-wide conversation about homelessness.
Tory and the city were under pressure to call a cold weather alert as the temperature continued to drop, which automatically opened more shelter space in the city.
On Tuesday, Tory opened warming centres in the city and the cold weather alert was eventually issued.