This is what we found on Wikipedia regarding the Leith Comedy Festival:

This is a list of arts and cultural festivals regularly taking place in Edinburgh, Scotland.

The city has become known for its festivals since the establishment in 1947 of the Edinburgh International Festival and the Edinburgh Festival Fringe which runs alongside it. The latter is the largest event of its kind in the world.

The term Edinburgh Festival is commonly used, but there is no single festival; the various festivals are put on by separate, unrelated organisations. However they are widely regarded as part of the same event, particularly the various festivals that take place simultaneously in August each year. The term Edinburgh Festival is often used to refer more specifically to the Fringe, being the largest of the festivals; or sometimes to the International Festival, being the original "official" arts festival. Within the industry, people refer to all the festivals collectively as the Edinburgh Festivals (plural).