the chronological future/past of a Cauchy surface $\Sigma$ of $M$. Unitality of the product follows immediately from the fact that there exists a unique empty tuple $\emptyset\to N$ for each $N\in\Loc$ and the composition properties of prefactorization algebras.

Naturality of algebra structures
--------------------------------

The aim of this subsection is to investigate compatibility between the algebra structures from Proposition \[propo:associativeunital\] and the maps $\FFF(f): \FFF(M)\to\FFF(N)$ induced by $\Loc$-morphisms. For our main statement to be true it will be crucial to assume that $\FFF \in \PFA^{\addc}$ is a Cauchy constant additive prefactorization algebra in the sense of Definitions \[def:additivePFA\] and \[def:CauchyconstantPFA\]. As a first partial result, we prove the following general statement.

\[lem:RCmultcompatibility\] Let $\FFF\in\PFA^c$ be any Cauchy constant prefactorization algebra (not necessarily additive). Let further $f:M\to N$ be a $\Loc$-morphism such that the image $f(M)\subseteq N$ is relatively compact. Then $\FFF(f) : \FFF(M)\to\FFF(N)$ preserves the multiplications and units from Proposition \[propo:associativeunital\], i.e.$\mu_N\,(\FFF(f)\otimes\FFF(f)) = \FFF(f)\,\mu_M$ and $\eta_N = \FFF(f)\, \eta_M$.

The units are clearly preserved for every $\Loc$-morphism $f:M\to N$ because composing the unique empty tuple $\emptyset \to M$ with $f:M\to N$ yields the unique empty tuple $\emptyset\to N$.

Let us focus now on the multiplications. Because $f(M)\subseteq N$ is by hypothesis 