j$. We shall write ${\underline{f}} : {\underline{M}} \to N$ for tuples ${\underline{f}} = (f_1,\dots,f_n)$ of pairwise disjoint morphisms.

By convention, a $1$-tuple ${\underline{f}} =(f): {\underline{M}}\to N$ of pairwise disjoint morphisms is just a $\Loc$-morphism $f:M\to N$ and there exists a unique empty tuple $\emptyset\to N$ for each $N\in \Loc$.

\[subsec:PFA\]Factorization algebras
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Factorization algebras are typically considered in the context of topological, complex or Riemannian manifolds, see [@CostelloGwilliam] for a detailed study. In order to obtain a meaningful comparison to algebraic quantum field theory, which is typically considered in the context of globally hyperbolic Lorentzian manifolds, we shall introduce below a variant of factorization algebras on the category $\Loc$ from Definition \[def:Loc\]. A similar concept of factorization algebras on $\Loc$ appeared before in [@GwilliamRejzner]. For what follows let us fix any cocomplete closed symmetric monoidal category $(\CC,\otimes,I,\tau)$, e.g. the category of vector spaces $\Vec_\bbK^{}$ over a field $\bbK$.

A [*prefactorization algebra*]{} $\FFF$ on $\Loc$ with values in $\CC$ is given by the following data:

-   for each $M\in \Loc$, an object $\FFF(M)\in\CC$;

-   for each tuple ${\underline{f}}=(f_1,\dots,f_n) : {\underline{M}} \to N$ of pairwise disjoint morphisms, a $\CC$-morphism $\FFF({\underline{f}}) : \bigotimes_{i=1}^n \FFF(M_i)\to \FFF(N)$ (called [*factorization product*]{}), with the convention that to 