with values in $\CC$ is a functor $\AAA : \Loc\to \Alg$ that satisfies the Einstein causality axiom: for every pair of causally disjoint morphisms $(f_1: M_1\to N)\perp (f_2:M_2\to N)$, the diagram

\[eqn:Einsteincausality\] @C=5em[ \_-[(f\_1)(f\_2)]{} (M\_1)(M\_2)\^-[(f\_1)(f\_2)]{} & (N)(N)\^-[\_N\^]{}\
(N)(N)\_-[\_N]{}& (N) ]{}

in $\CC$ commutes, where $\mu_N^{(\op)}$ denotes the (opposite) multiplication on $\AAA(N)$. A morphism $\kappa : \AAA\to \BBB$ of algebraic quantum field theories is a natural transformation between the underlying functors.

We denote by $\AQFT$ the category of algebraic quantum field theories on $\Loc$.

For proving some of the results of this paper, we require a relatively mild variant of an additivity property in the sense of [@Fewster]. Recall from Definition \[def:RCM\] the category $\RC_M$ of relatively compact and causally convex open subsets of $M\in\Loc$.

\[def:AQFTadditive\] An algebraic quantum field theory $\AAA\in\AQFT$ is called [*additive*]{} if, for every $M\in\Loc$, the canonical morphism

@C=3em[ (\_M : \_M) \^- & (M) ]{}

is an isomorphism in $\Alg$. We denote by $\AQFT^{\add}\subseteq \AQFT$ the full subcategory of additive algebraic quantum field theories.

\[rem:algcolim\] Because $\RC_M$ is a directed set by Lemma \[lem:RCMdirected\], the colimit in Definition \[def:AQFTadditive\] can be computed in the underlying category $\CC$, see e.g. [@Fresse Proposition 1.3.6]. Hence, to check if 