paper does not study this generalization and will focus on the case of $1$-categorical quantum field theories.) We would like to state very clearly that our results prove an equivalence theorem between certain categories of prefactorization algebras and algebraic quantum field theories, hence they [*do not*]{} make any statements about the relationship between explicit construction methods for examples. We refer to [@GwilliamRejzner] for a concrete comparison between BV quantization [@CostelloGwilliam] and perturbative canonical quantization [@FredenhagenRejzner; @Rejzner].

Let us now explain in more detail our constructions and results while outlining the content of the present paper: In Section \[sec:prel\] we recall the necessary preliminaries from Lorentzian geometry, factorization algebras and algebraic quantum field theory. All prefactorization algebras and algebraic quantum field theories will be defined on the usual category $\Loc$ of oriented and time-oriented globally hyperbolic Lorentzian manifolds. We introduce an additivity axiom for both prefactorization algebras and algebraic quantum field theories, which roughly speaking demands that the observables in a spacetime $M$ are generated by the observables in the [*relatively compact*]{} and causally convex open subsets $U\subseteq M$. It is shown that factorization algebras, i.e.prefactorization algebras satisfying Weiss descent, are in particular additive prefactorization algebras. We also introduce a Cauchy 