]{}

where the upward-left pointing arrow is an isomorphism because $\FFF$ is by hypothesis Cauchy constant. A priori, it is not clear whether different choices of such $\iota_{{\underline{U}}}^M : {\underline{U}}\to M$ lead to the same multiplication map in . The possible choices are recorded in the following category.

\[def:PM\] Let $M\in\Loc$. We denote by $\PP_M$ the category whose objects are all pairs of disjoint morphisms $\iota_{{\underline{U}}}^M = (\iota_{U_+}^M,\iota_{U_-}^M): {\underline{U}}\to M$ corresponding to causally convex open subsets $U_+,U_-\subseteq M$ that satisfy

-   there exists a Cauchy surface $\Sigma$ of $M$ such that $U_\pm \subseteq I^\pm_M(\Sigma)$, and

-   $\iota_{U_\pm}^M: U_\pm\stackrel{c}{\to} M$ are Cauchy morphisms.

There exists a unique morphism $(\iota_{{\underline{U}}}^M :  {\underline{U}}\to M) \to (\iota_{{\underline{V}}}^M :  {\underline{V}}\to M)$ if and only if $U_\pm\subseteq V_\pm$.

\[lem:PPMconnected\] For every $M\in\Loc$, the category $\PP_M$ is non-empty and connected.

[*Non-empty:*]{} Choose any Cauchy surface $\Sigma$ of $M$ and define $\Sigma_\pm := I^\pm_M(\Sigma)$. Then $\iota_{{\underline{\Sigma}}}^M = (\iota_{\Sigma_+}^M,\iota_{\Sigma_-}^M) : {\underline{\Sigma}}\to M$ defines an object in $\PP_M$.

[*Connected:*]{} We have to prove that there exists a zig-zag of morphisms in $\PP_M$ between every pair of objects $\iota_{{\underline{U}}}^M :  {\underline{U}}\to M$ and $\iota_{{\underline{V}}}^M :  {\underline{V}}\to M$. For every object $\iota_{{\underline{U}}}^M: {\underline{U}}\to M$ in $\PP_M$, there exists by hypothesis a Cauchy 