Is the inclusion of the category of sheaves into the category of presheaves monadic? If not, then maybe it preserves directed colimits?

-
Ok. I think the answer is no. Sorry for disturbing. –  thinker Dec 18 '11 at 16:17
The answer to the second question is "not in general", but it is true that it will always be accessible, i.e. preserve sufficiently-highly-filtered colimits. For some purposes that is good enough. –  Mike Shulman Dec 20 '11 at 1:17