sum up those multiplicities.

\[ex:global-annot\] Given $n\in\mathbb{N}\backslash\{0\}$, we define the functor $\mathcal{B}^n:\GR \to \Mon$: For every graph $G$, $\mathcal{B}^n(G) = \mathcal{M}_n^{V_G\cup E_G}$. For every graph morphism $\varphi \colon G \to H$ and $a \in \mathcal{B}^n(G)$, we have $\mathcal{B}^n_{\varphi}(a) \in \mathcal{M}_n^{V_{H} \cup E_{H}}$ with: $$\mathcal{B}^n_{\varphi}(a)(y) = \sum\limits_{\varphi(x)=y}^{}
    a(x), \quad 
    \textit{where } x \in (V_G \cup E_G) \textit{ and } y \in (V_{H} \cup 
    E_{H}).$$ Therefore an annotation based on a functor $\mathcal{B}^n$ associates every item of a graph with a number (or the top value $*$). We will call such annotations *multiplicities*. Furthermore the action of the functor on a morphism transforms a multiplicity by summing up (in $\mathcal{M}_n$) the values of all items of the source graph that are mapped to the same item of the target graph.

For a graph $G$, its *standard multiplicity* $s_G \in \mathcal{B}^n(G)$ is defined as the function which maps every node and edge of $G$ to $1$.

As another example we consider local annotations which record the out-degree of a node and where the action of the functor is to take the supremum instead of the sum.

\[ex:local-annot\] Given $n\in\mathbb{N}\backslash\{0\}$, we define the functor $\mathcal{S}^n:\GR \to \Mon$ as follows: For every graph $G$, $\mathcal{S}^n(G) 