Let $C$ be an acyclic category with a terminal object $t$, in "Combinatorial algebraic topology" by Kozlov he defines Mobius functions for acyclic categories, he first starts by defining a function $\mu: \mathcal{O}(C) \rightarrow \mathbb{Z}$ as follows (page 174, link: http://www.maths.ed.ac.uk/~aar/papers/kozlov.pdf):
$\mu (t) = 1;$
for $x \in \mathcal{O}(C)$, $x \neq t$, we set
$\mu (x) := - \sum\limits_{m} \mu (\partial_{\bullet} m) $
where the sum is taken over all nonidentity morphisms $m \in \mathcal{M}(C)$ such that $\partial^{\bullet} m = x$.
Here $\mathcal{O}(C)$ and $\mathcal{M}(C)$ are respectively the set of objects and morphisms of $C$, and for a morphism $f:a \rightarrow b$ Kozlov writes $\partial^{\bullet} f = dom \, f = a$ and $\partial_{\bullet} f = cod \, f = b$.
My question is:
What is the value of $\mu(\partial_{\bullet}m)$ for a morphism $m: x \rightarrow y$ when $x, y \neq t$? Or what am I missing here?
Maybe am I missing something, but without knowing the acyclic category in question, the answer is $\mu(\partial_{\bullet} m) = \mu(y)$.
Notice that $\mu$ is a recursive function.
For example, suppose that we are asked to calculate the value of $\mu(a)$ for the category $a \to b \to t$. Then, since we have morphisms $a \to b$ and $a \to t$, first we need to calculate $\mu(b)$ and $\mu(t)$.