If you do that, the yeast will clean up and you'll get less yeast character. It's not so much a Fullers thing as a fairly general British approach - pitch quite cool (15-16°C), let it free-rise (at homebrew scale this may need a little external heat) to say 20°C at high krausen, then back down to 17-18°C to complete fermentation, then condition at cellar temperature.
There's some variations on that - some like Morland go as high as 24°C - but that's the general idea. Whilst going warmer encourages esters, too long at high temps encourages the yeast to clean them up.
It's a bit different when you're packaging into cask rather than bottle, bottle conditioning is fairly rare in the UK as retailers hate it although some brewers do it for one premium bottle as a nod to CAMRA.