prefactorization algebras from Section \[sec:inverseconstruction\] is better suited to the category of Lorentzian spacetimes $\Loc$ than the more naive concept from Section \[subsec:PFA\] that allows also for factorization products for non-time-orderable tuples of pairwise disjoint morphisms. In Section \[sec:equivalence\] we explain that the construction $\bbA : \PFA^{\addc}\to\AQFT^{\addc}$ from Section \[sec:construction\] factors through the forgetful functor $\PFA^{\addc}\to \toPFA^{\addc}$, thereby defining a functor $\bbA: \toPFA^{\addc}\to\AQFT^{\addc}$ that assigns a Cauchy constant additive algebraic quantum field theory to each Cauchy constant additive [*time-orderable*]{} prefactorization algebra. Our main Equivalence Theorem \[theo:equivalence\] proves that this functor admits an inverse that is given by the restriction $\bbF : \AQFT^{\addc}\to\toPFA^{\addc}$ of the functor from Section \[sec:inverseconstruction\] to Cauchy constant and additive theories. Hence, Cauchy constant additive algebraic quantum field theories are naturally identified with Cauchy constant additive time-orderable prefactorization algebras. In Section \[subsec:Involutions\], we use our main Equivalence Theorem \[theo:equivalence\] to transfer $\ast$-involutions from algebraic quantum field theories to Cauchy constant additive time-orderable prefactorization algebras. By construction, we obtain an equivalence $\ast\AQFT^{\addc}\simeq \ast\toPFA^{\addc}$ between theories with $\ast$-involutions. We show that the transferred concept of $\ast$-involutions for Cauchy constant additive time-orderable prefactorization algebras involves Cauchy constancy explicitly, hence it does not extend to generic time-orderable prefactorization algebras in $\toPFA$. 