Interpreting $\FFF(M)$ as a collection of observables for a quantum field theory, this means that all observables described by $\FFF(M)$ arise from relatively compact regions $U\subseteq M$.

Every factorization algebra $\FFF$ on $\Loc$ is an additive prefactorization algebra.

Suppose that $\FFF$ is a factorization algebra [@CostelloGwilliam], i.e. it satisfies a cosheaf condition with respect to all Weiss covers of every $M\in \Loc$. For every $M\in\Loc$, the cover defined by $\RC_M$ is a Weiss cover. Indeed, given finitely many points $x_1,\dots,x_n\in M$, there exist $U_i\in\RC_M$ with $x_i\in U_i$ and hence $U\in\RC_M$ with $x_1,\dots,x_n\in U$ because $\RC_M$ is directed by Lemma \[lem:RCMdirected\]. The property of being a factorization algebra then implies that the canonical diagram

\[eqn:Cechfacalg\] @C=3em[ \_ (UV) @&lt;0.5ex&gt;\[r\]@&lt;-0.5ex&gt;\[r\] & \_[U\_M]{} (U)  & (M) ]{}

is a coequalizer in $\CC$. Our claim then follows by observing that the cocones of are canonically identified with the cocones of . Indeed, any cocone $\{\alpha_U : \FFF(U) \to Z\}$ of defines a cocone of because $U\cap V \in\RC_M$ (whenever nonempty) and hence the diagram

@R=1em@C=3em[ &(U)\^-[\_U]{}&\
(UV)\^-[\_[UV]{}]{}\_-[(\_[UV]{}\^V)  ]{}\^-[(\_[UV]{}\^U)  ]{}&&Z\
&(V)\_-[\_V]{}& ]{}

in $\CC$ commutes. Vice versa, any cocone $\{\alpha_U : \FFF(U) \to Z\}$ of defines a cocone of because $U\cap V =U$, for all 