TITLE: Categorical semantics of W-types
QUESTION [9 upvotes]: Jacob's book titled "Categorical Logic and Type Theory" gives a nice description of Π and Σ types as adjunctions to substitution functors induced by display maps.  Is there a similar categorical description of W-types (and maybe M-types while we are at it)?

REPLY [3 votes]: Also, this may be helpfull —
http://ncatlab.org/nlab/show/polynomial+functor
http://ncatlab.org/nlab/show/W-type