Categorification is a bit like quantization: it isn't a construction so much as a desideratum for a relationship between one thing and another (in the case of categorification an (n+1)-categorical structure and an n-categorical structure; in the case of quantization a quantum mechanical system and a classical mechanical system). Categorification wants to find a higher-dimensional categorical structure corresponding to a lower-dimensional one by weakening equations to natural isomorphisms and imposing new, sensible, coherence conditions. In general, for the original purpose for which it was proposed--constructions of TQFT's and models of quantum gravity--one wants the highest categorical level to have a linear structure (hence Baez wanting tensor product and a sum it distributes over, rather than cartesian product and coproduct). Specific lower-dimensional categories with structure are 'categorified' by finding a higher-dimensional category with the new structure which 'lies over' the lower dimensional one in the way an additive monoidal category lies over its Grothendieck rig. For instance any (k-linear) monoidal category with monoid of isomorphism classes M is a categorification of M, and more generally (k-linear) monoidal categories are a categorification of monoids. A simple example shows why it is not a construction: commutative monoids (as rather special categories with one object) admit two different categorifications: symmetric monoidal categories and braided monoidal categories (each regarded as a kind of bicategory with one object). There is a good argument for regarding braided monoidal categories as the 'correct' categorification: the Eckmann-Hilton theorem ('a group in GROUPS is an abelian group' or, really as the proof shows, 'a monoid in MONOIDS is a commutative monoid') 'categorifies' to: A monoidal category in MONCAT is a braided monoidal category.