constancy (or time-slice) axiom for both kinds of theories, which formalizes a concept of time evolution in a globally hyperbolic Lorentzian manifold. In Section \[sec:construction\] we construct a functor $\bbA: \PFA^{\addc}\to\AQFT^{\addc}$ that assigns a Cauchy constant additive algebraic quantum field theory $\bbA[\FFF]$ to each Cauchy constant additive prefactorization algebra $\FFF$, see Theorem \[theo:PFAtoAQFT\] for the main result. The crucial step is to define canonical multiplications $\mu_M : \FFF(M)\otimes \FFF(M)\to\FFF(M)$ for such $\FFF$ (cf. ), which is done by using Cauchy constancy. Proving naturality and Einstein causality of these multiplications requires the additivity axiom, cf. Propositions \[propo:multnaturality\] and \[propo:caus\]. In Section \[sec:inverseconstruction\] we construct a functor $\bbF : \AQFT\to\toPFA$ that assigns a [*time-orderable*]{} prefactorization algebra $\bbF[\AAA]$ to each algebraic quantum field theory $\AAA$, see Theorem \[theo:AQFTtotPFA\] for the main result. The difference between time-orderable and ordinary prefactorization algebras on $\Loc$ is that the former just encode factorization products $\FFF({\underline{f}}) : \bigotimes_{i=1}^n \FFF(M_i)\to N$ for tuples of pairwise disjoint morphisms ${\underline{f}}$ that are in a suitable sense time-orderable, see Definition \[def:timeordered\]. There is a natural forgetful functor $\PFA\to\toPFA$ from ordinary to time-orderable prefactorization algebras, which is however not full, see Remarks \[rem:nontimeorderable\] and \[rem:tPFAvsPFA\]. Our results suggest that the concept of time-orderable 