Notions of M\"obius inversion

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.