TITLE: univalent axiom as a property of a model category? 
QUESTION [5 upvotes]: I am interested to understand the univalence axiom of Voevodsky; however, I know
very little type theory. Thanks to response below, I now understand what is being univalent
means for a morphism. A couple more questions, though:

Do I understand correctly that tthe Univalence Axiom makes sense for an arbitrary locally cartesian closed model category ? And what is 'the universe of small fibrations'? If I understand correctly, it means that for each small fibration g:Y→X there are morphisms h:X→U and h˜:Y→U˜ such that the corresponding square is a pull-back square. What does exactly 'small' mean here?
 Is there a
reformulation of the univalence axiom
stated fully in terms of a model
category, perhaps with an
distinguished fibration ?  What is the best reference giving full detail?

I was not able to find anything in the literature.
The NSF proposal of Voevodsky seem to come  quite close to giving such a formulation,
but it does not have full details.

REPLY [2 votes]: This note in arxiv:1111.3489 gives an formal interpretation of the univalence axiom in an arbitrary (locally cartesian closed) model category: basically, interpret word-by-word a passage of Voevodsky
describing the univalence for sSets. 
 The authors are non-specialist, though, so various blunders possible.