algebras are prefactorization algebras that satisfy a suitable descent condition with respect to Weiss covers [@CostelloGwilliam]. For proving our results in this paper, it is sufficient to assume a weaker descent condition that we shall call [*additivity*]{} in reference to a similar property in algebraic quantum field theory [@Fewster]. As explained below, this includes in particular all factorization algebras on $\Loc$. Before we can formalize the additivity property, we have to introduce some further terminology and notations.

\[def:RCM\] For $M\in\Loc$, we denote by $\RC_M$ the [*category of all relatively compact and causally convex open subsets*]{} $U\subseteq M$ with morphisms given by subset inclusions.

Note that the assignment $M \mapsto \RC_M$ may be promoted to a functor $\RC_{(-)} : \Loc\to \Cat$ with values in the category of (small) categories. Concretely, given any $\Loc$-morphism $f:M\to N$, then the functor $\RC_f : \RC_M\to\RC_N$ sends each relatively compact and causally convex open subset $U\subseteq M$ to its image $f(U)\subseteq N$. Since $f$ is continuous, it follows that this is a relatively compact and causally convex open subset of $N$.

\[lem:RCMdirected\] For every $M\in\Loc$, the category $\RC_M$ is a directed set.

Let $U_1, U_2 \in \RC_M$. We shall construct $U \in \RC_M$ such that $U_i 