recall that there exists a unique empty tuple of disjoint morphisms $\emptyset\to M$ to which the prefactorization algebra assigns a $\CC$-morphism that we shall denote by $\eta_M : I\to \FFF(M)$. The main result of this subsection is as follows.

\[propo:associativeunital\] Let $\FFF\in\PFA^c$ be any Cauchy constant prefactorization algebra. For every $M\in\Loc$, the object $\FFF(M)\in\CC$ carries the structure of an associative and unital algebra in $\CC$ with multiplication $\mu_M : \FFF(M)\otimes\FFF(M)\to \FFF(M)$ given by and unit $\eta_M : I\to \FFF(M)$ given by evaluating $\FFF$ on the empty tuple $\emptyset \to M$.

To prove that the multiplication $\mu_M$ is associative, we consider two Cauchy surfaces $\Sigma_0,\Sigma_1$ of $M$ such that $\Sigma_1 \subset I^+_M(\Sigma_0)$, i.e.$\Sigma_1$ is in the future of $\Sigma_0$. Using the independence result from Corollary \[cor:multunique\] and the composition properties of prefactorization algebras from Section \[subsec:PFA\], one easily confirms that $\mu_M\, (\id\otimes \mu_M)$ is the upper path and $\mu_M\, (\mu_M\otimes \id )$ the lower path from $\FFF(M)^{\otimes 3}$ to $\FFF(M)$ in the commutative diagram

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(M)(M)(M) && (M)\
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where as before we denote by $\Sigma_{\pm } := I^\pm_M(\Sigma) \subseteq M$ 