TITLE: Some reference for categorical logic?
QUESTION [6 upvotes]: By "categorical logic" I mean category-theoretical models of logic. In particular, I am more interested in models of intuitionistic predicate logic with conjunction, disjunction, implication and quantifiers.
I know about a book of Lambek and Scott and another one by Makkai and Reyes, but these are quite old, especially the second. Does anyone know any newer books/resources on this topic?
For example, I have studied on propositional intuitionistic logic (with $\bot,\wedge,\vee$ and $\to$) and the corresponding category-theoretical model is a bicartesian closed category (objects are formulas and arrows are proofs). But what is the case with predicate intuitionistic logic? Is there any more recent text analyzing this case?

REPLY [7 votes]: Besides the very good books on topos theory already mentioned, I highly recommend Categorical Logic by Andrew Pitts for an introduction to the categorical semantics of type theories (you can find it online here. See Section 5 for the treatment of predicate logic).
The more advanced Categorical Logic and Type Theory by B. Jacobs, from the series Studies in Logic and the Foundations of Mathematics, is another good reference on the subject.