that the image of the $\Loc$-morphism $f: M\to N$ is relatively compact, then it is [*not*]{} true that the image $f(\Sigma)\subset N$ of a Cauchy surface $\Sigma$ of $M$ can be extended to a Cauchy surface $\widetilde{\Sigma}$ of $N$. A simple example that demonstrates this feature is given by the subset inclusion $\iota_{U}^V : U\to V$ of the following two diamond regions in $2$-dimensional Minkowski spacetime (note that $U$ is not relatively compact as a subset of $V$):

\[eqn:nonrelativelycompact\]

(-1,0) – (0,1) – (1,0) – (0,-1) – (-1,0); (-0.5,0.5) – (0,1) – (0.5,0.5) – (0,0) – (-0.5,0.5); (0,-0.5) node [[$V$]{}]{}; (0,0.75) node [[$U$]{}]{}; (-0.5,0.5) .. controls (0,0.55) .. (0.5, 0.5) node\[midway, below\] [[$\Sigma$]{}]{}; (-1.25,-0.8) – (-1.25,0.8) node\[above\] [[time]{}]{};

It is evident that no Cauchy surface $\Sigma$ of $U$ admits an extension to a Cauchy surface of $V$. Hence, $\FFF(\iota_{U}^V) : \FFF(U)\to \FFF(V)$ may fail to preserve the multiplications. We shall show below that the issues explained in this remark are solved by considering [*additive*]{} prefactorization algebras as in Definition \[def:additivePFA\].

The main result of this subsection is as follows.

\[propo:multnaturality\] Let $\FFF\in\PFA^{\addc}$ be any Cauchy constant additive prefactorization algebra. For every $\Loc$-morphism $f:M\to N$, the $\CC$-morphism $\FFF(f) : \FFF(M)\to\FFF(N)$ preserves 