  M\"obius inversion, originally a tool in number theory, was generalized to
posets for use in group theory and combinatorics. It was later generalized to
categories in two different ways, both of which are useful. We provide a
unifying abstract framework. This allows us to compare and contrast the two
theories of M\"obius inversion for categories, and advance each of them. Among
several side benefits is an improved understanding of the following fact: the
Euler characteristic of the classifying space of a (suitably finite) category
depends only on its underlying graph.
