
The pair replace Dominic Mohan, who was made the editor of the newspaper last month following the departure of Rebekah Brooks (née Wade).
Brooks, who had been the editor of The Sun for more than six years, left to take on the position of chief executive at the newspaper's parent company, News International.
Cosyns and Webster have both been working at The Sun since 2003.
Cosyns joined the newspaper from the Daily Mirror, where he had held the position of art director, while Webster joined The Sun from the News of the World, where he was the head of pictures.