In Tom Leinster, The Euler Characteristic Of A Category, the author generalizes the notion of Möbius Inversion for posets to finite categories. This violates the principle of equivalence. A possible solution i found to that would be to consider essentially finite categories and apply the definitions given to the skeleton of such a category (the Möbius Inversion only exists for skeletal categories anyway according to the text). Since the definition of mobius inversion ignores compositions of morphisms, the author remarks:
The definitions above could be made for directed graphs rather than categories, since they do not refer to composition. However, this generality seems to be inappropriate. For example, the definition of Möbius inversion will lead to a definition of Euler characteristic, and if we use graphs rather than categories then we obtain something other than ‘vertices minus edges’. Proposition 2.10 clarifies this point. (...)
Proposition 2.10 Let G be a finite circuit-free directed graph. Then $\chi(F(G))$ is defined and equal to $|G_0|-|G_1|$.
Here $G_0$ and $G_1$ are the sets of vertices and edges respectively, and $F(G)$ is the free category on $G$. I don't understand the authors explanation for why to define this for finite skeletal categories instead of finite circuit-free directed graphs, nor how Proposition 2.10 relates to that. It seems inappropriate to me to define it for categories since this notion is not preserved for equivalence of categories, but it is preserved for isomorphisms of graphs. Am i missing something?