The term "cognitive function" refers to an aspect of thought process, while "affect" refers to the more visceral, emotional and gut felt aspects of the psyche. The feeling functions bare strong resemblances to affective phenomena. Both are involved with evaluation and determining what is basically good or bad.
For instance: "The introvert of feeling-type finds support and guidance by shaping his own feeling-attitudes in accordance with an inner ideal. Here the activities of feeling are hidden, and from the outside there is, as a rule, little to tell us that we are dealing with a person of feeling-type. Feeling aims more especially at an inner harmony, trying to discover what under various circumstances should be the right relationships between people if life is to be beautiful and well balanced. Reality, however, reveals in most cases that this ideal is not attained, and introverted feeling is particularly vulnerable in regard to such experiences. This vulnerability — which may become as intense at that of the sensitive plant — is one of the most characteristic peculiarities of this type. "
--Dr. J. H. van der Hoop, Lecturer in Psychiatry, Amsterdam, Conscious Orientation
However, it seems that maybe there is some confusion as to where emotion should be sorted into. Should it only be headed under affect, or does it have a place in cognition as well? Both cognition and affect probably influence each other. They come before and after each other, depending on the impetus that the environment poses on the subject.
My opinion is that emotions aren't cognitive, but can be rationalized and understood cognitively. Cognitive functions, and maybe Feeling functions in particular, play a role in the rationalization of affect.