of $M$. To prove the last statement, consider any inextensible time-like curve $\gamma : I\to M$, which we may assume without loss of generality to be future directed. (If $\gamma$ would be past directed, then change the orientation of the interval $I$.) Because $\Sigma_1$ and $\Sigma_1^\prime$ are Cauchy surfaces of $M$, there exist [*unique*]{} $t,t^\prime\in I$ such that $\gamma(t)\in\Sigma_1$ and $\gamma(t^\prime)\in\Sigma_1^\prime$. If $t\geq t^\prime$, then $\gamma(t)\in \Sigma_1 \cap J^+_M(\Sigma_1^\prime)\subseteq \widetilde{\Sigma}$, and if $t^\prime \geq t$, then $\gamma(t^\prime) \in J^+_M(\Sigma_1)\cap 
\Sigma_1^\prime\subseteq \widetilde{\Sigma}$. Hence, $\gamma$ meets $\widetilde{\Sigma}\subset M$ at least once. Multiple intersections are excluded by the definition of $\widetilde{\Sigma}$ in and the fact that both $\Sigma_1$ and $\Sigma_1^\prime$ are Cauchy surfaces of $M$.

\[cor:multunique\] For every $M\in\Loc$, the multiplication map $\mu_M$ in does not depend on the choice of object $\iota_{{\underline{U}}}^M : {\underline{U}}\to M$ in $\PP_M$.

By Lemma \[lem:PPMconnected\], it is sufficient to prove that $\iota_{{\underline{U}}}^M : {\underline{U}}\to M$ and $\iota_{{\underline{V}}}^M : {\underline{V}}\to M$ define the same multiplication if $U_+ \subseteq V_+$ and $U_-\subseteq V_-$. This is a consequence of the commutative diagram

@C=4em[ &\^-\_-[(\_[V\_+]{}\^M)(\_[V\_-]{}\^M)     ]{}(V\_+)(V\_-)\^-[(\_\^M)]{} &\
(M)(M) && (M)\
&\_-\^-[(\_[U\_+]{}\^M)(\_[U\_-]{}\^M)     ]{} (U\_+)(U\_-)\_-[(\_\^M)]{}\_-[(\_[U\_+]{}\^[V\_+]{})(\_[U\_-]{}\^[V\_-]{})]{}& ]{}

where one also uses the composition properties of prefactorization algebras.

To obtain a unit for $\FFF(M)$, we 