to each spacetime $M$ a [*vector space*]{} $\FFF(M)$ of observables and to each tuple ${\underline{f}} = (f_1:M_1\to N,\dots,f_n: M_n\to N)$ of pairwise disjoint spacetime embeddings a factorization product $\FFF({\underline{f}}) : \bigotimes_{i=1}^n\FFF(M_i) \to \FFF(N)$ satisfying suitable properties, cf. [@CostelloGwilliam] and Section \[subsec:PFA\]. On the other hand, an algebraic quantum field theory $\AAA$ assigns to each spacetime $M$ an associative and unital [*$\ast$-algebra*]{} $\AAA(M)$ of observables and to each spacetime embedding $f: M\to N$ a $\ast$-algebra morphism $\AAA(f) : \AAA(M)\to \AAA(N)$ such that suitable axioms hold true, cf. [@Brunetti; @FewsterVerch; @AQFTbook; @BSWoperad] and Section \[subsec:AQFT\]. The main differences are that, in contrast to an algebraic quantum field theory $\AAA$, a prefactorization algebra $\FFF$ [*does not*]{} in general come endowed with 1.) a multiplication of observables in $\FFF(M)$, i.e. on the same spacetime $M$, because $(\id_M : M\to M,\id_M:M\to M)$ is not a pair of disjoint spacetime embeddings, and 2.) a concept of $\ast$-involution on observables in $\FFF(M)$.

In this paper we shall develop functorial constructions (cf. Theorems \[theo:PFAtoAQFT\] and \[theo:AQFTtotPFA\]) that allow us to relate prefactorization algebras and algebraic quantum field theories, provided that we assume certain natural hypotheses on both sides. We shall focus on the case where spacetimes are described 