relatively compact, its closure $\overline{f(M)}\subseteq N$ is compact. Let us take any Cauchy surface $\Sigma$ of $M$ and note that $\overline{f(\Sigma)} \subseteq N$ is a compact subset. Using further that $f(M)\subseteq N$ is causally convex and that the causality relation induced by time-like curves is open (cf. [@ONeill Lemma 14.3]), it follows that $\overline{f(\Sigma)} \subseteq N$ is achronal, i.e. every time-like curve in $N$ meets this subset at most once. By [@BernalSanchez Theorem 3.8], there exists a Cauchy surface $\widetilde{\Sigma}$ of $N$ such that $f(\Sigma) \subseteq \widetilde{\Sigma}$.

Using the Cauchy surfaces constructed above, we can define the multiplication $\mu_{M}$ in terms of $\Sigma_\pm := I^\pm_M(\Sigma)$ and the multiplication $\mu_N $ in terms of $\widetilde{\Sigma}_\pm := I^\pm_N(\widetilde{\Sigma})$, cf. . By construction, $f:M\to N$ restricts to $\Loc$-morphisms $f_{\Sigma_\pm}^{\widetilde{\Sigma}_\pm} : \Sigma_\pm \to \widetilde{\Sigma}_\pm$. Our claim that $\FFF(f) : \FFF(M)\to \FFF(N)$ preserves the multiplications then follows by observing that the diagram

@C=7em[ \_-[(f)(f)]{} (M)(M) & \^-\_-[(\_[\_+]{}\^M)(\_[\_-]{}\^M)]{} \^-[(f\_[\_+]{}\^[\_+]{} )(f\_[\_-]{}\^[\_-]{} )]{} (\_+)(\_-) \^-[(\_\^M)]{}& \^-[(f)]{} (M)\
(N)(N) & \_-\^-[(\_[\_+]{}\^N) (\_[\_-]{}\^N) ]{} (\_+)(\_-) \_-[(\_\^N)]{}& (N)\
]{}

commutes.

We would like to emphasize that our assumption that the image $f(M)\subseteq N$ is relatively compact was crucial for the proof of Lemma \[lem:RCmultcompatibility\]. In fact, if one [*does not*]{} assume 