TITLE: Functors preserve isomorphisms, sections and retracts. Do they reflect these properties?
QUESTION [6 upvotes]: Functors preserve isomorphisms, sections and retracts. Which of these properties, if any, do functors reflect?
Are there other fundamental properties preserved and or reflected by functors? 
Please specify covariant or contravariant if applicable.
Don't assume I understand any cat theory - all I know is triangles and functional languages.

REPLY [8 votes]: You can find a table of many preserved and reflected properties of functors in wonderful "The Joy of Cats" (JOC) book which you can find online at:
http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf
In general:

functors need not reflect isomorphisms (JOC 3.22)
functors need not reflect sections (JOC 7.23)
functors need not reflect retracts (JOC 7.29)

However: full and faithful functors reflect isomorphisms, sections and retracts (JOC 7.30)