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?