associative and unital algebra in $\CC$. This step relies on Cauchy constancy, while it does not require that the additivity property holds true. Step (2) consists of proving that these algebra structures are compatible with the maps $\FFF(f) : \FFF(M)\to\FFF(N)$ induced by $\Loc$-morphisms $f:M\to N$. Here our additivity property turns out to be crucial. Finally, in step (3) we show that the resulting functor $\Loc\to \Alg$ satisfies the properties of a Cauchy constant additive algebraic quantum field theory, cf. Section \[subsec:AQFT\].

Object-wise algebra structure
-----------------------------

All results of this subsection do not use the additivity property from Definition \[def:additivePFA\]. Hence, we let $\FFF\in \PFA^{c} $ be any Cauchy constant prefactorization algebra.

Let us fix any $M\in\Loc$. The basic idea to define a multiplication map $\mu_M : \FFF(M)\otimes\FFF(M)\to \FFF(M)$ is as follows: Consider two causally convex open subsets $U_+,U_-\subseteq M$ satisfying (i) there exists a Cauchy surface $\Sigma$ of $M$ such that $U_\pm \subseteq I^\pm_M(\Sigma)$, and (ii) $\iota_{U_\pm}^M: U_\pm\stackrel{c}{\to} M$ are Cauchy morphisms. In particular, $U_+\cap U_-=\emptyset$ are disjoint and hence provide a pair of disjoint morphisms $\iota_{{\underline{U}}}^M = (\iota_{U_+}^M,\iota_{U_-}^M) : {\underline{U}} \to M $. We define $\mu_M$ by the commutative diagram

\[eqn:multiplicationmap\] @C=3em[ (M)(M) \^-[\_M]{} && (M)\
&\_-\^-[(\_[U\_+]{}\^M)(\_[U\_-]{}\^M)     ]{} (U\_+)(U\_-)\_-[(\_\^M)]{}& 