On descent for coalgebras and type transformations

We find a criterion for a morphism of coalgebras over a Barrexactcategory to be effective descent and determine (effective) descentmorphisms for coalgebras over toposes in some cases. Also, we studysome exactness properties of endofunctors of arbitrary categories inconnection with natural transformations between them as well asthose of functors that these transformations induce betweencorresponding categories of coalgebras.  As a result, we findconditions under which the induced functors preserve natural numberobjects as well as a criterion for them to be exact. Also thisenable us to give a criterion for split epis in a category ofcoalgebras to be effective descent.