General idea: take a thing we know and add structure to it such that things that were formerly properties become structures
Do in such a way as to be able to recover the thing we categorified by forgetting the new structure.
In categorified world, we have more structure available to talk about the relationships between objects.
An example is how we treated preorders as categories.
Ordinary categories are Set-categories
Categorification of arithmetic using the category FinSet
Replace natural numbers with arbtirary sets of that cardinality.
Replace \(+\) with coproduct.
This is good categorification because, with replacements, arithmatic truths are preserved: \(\bar{5}\sqcup \bar{3} \cong \bar{8}\)