curve in $M$.

The following category of Lorentzian manifolds plays a fundamental role in algebraic quantum field theory, see e.g. [@Brunetti; @FewsterVerch; @AQFTbook; @BSWoperad].

\[def:Loc\] We denote by $\Loc$ the category whose objects are all oriented and time-oriented globally hyperbolic Lorentzian manifolds $M$ and morphisms are all orientation and time-orientation preserving isometric embeddings $f : M\to N$ with causally convex and open image $f(M)\subseteq N$.

We introduce the following terminology to specify important (tuples of) $\Loc$-morphisms that enter the definitions of algebraic quantum field theories and factorization algebras.

\[def:Locmaps\]

-   A $\Loc$-morphism $f : M\to N$ is called a [*Cauchy morphism*]{} if its image $f(M)\subseteq N$ contains a Cauchy surface of $N$. We shall write $f : M\stackrel{c}{\to} N$ for Cauchy morphisms.

-   A pair of $\Loc$-morphisms $(f_1:  M_1\to N,f_2: M_2\to N)$ to a common target is called [*causally disjoint*]{} if the images $f_1(M_1)\subseteq N$ and $f_{2}(M_2)\subseteq N$ are causally disjoint subsets of $N$. We shall write $f_1\perp f_2$ for causally disjoint morphisms.

-   A tuple of $\Loc$-morphisms $(f_1: M_1\to N,\dots,f_n : M_n\to N)$ to a common target is called [*pairwise disjoint*]{} if the images $f_i(M_i)\subseteq N$ are pairwise disjoint subsets of $N$, i.e. $f_i(M_i)\cap f_j(M_j) =\emptyset$, for all $i\neq 