$U\subseteq V$, and hence the diagram

@R=1em@C=3em[ &(U)\^-[\_U]{}\^-[(\_U\^V)]{}&\
(UV)\_-[(\_[UV]{}\^V)  ]{}@[=]{}\[ur\]&&Z\
&(V)\_-[\_V]{}& ]{}

in $\CC$ commutes.

As a last definition, we would like to introduce a suitable local constancy property that is adapted to the category $\Loc$. This property will play a crucial role in establishing our comparison results. Recall from Definition \[def:Locmaps\] the concept of Cauchy morphisms.

\[def:CauchyconstantPFA\] A prefactorization algebra $\FFF\in\PFA$ is called [*Cauchy constant*]{} if $\FFF(f) : \FFF(M)\to\FFF(N)$ is an isomorphism in $\CC$, for every Cauchy morphism $f : M\stackrel{c}{\to} N$. We denote by $\PFA^c\subseteq \PFA$ the full subcategory of Cauchy constant prefactorization algebras. The full subcategory $\PFA^{\addc}\subseteq \PFA^\add$ of Cauchy constant additive prefactorization algebras is defined analogously.

\[subsec:AQFT\]Algebraic quantum field theories
-----------------------------------------------

Let $\CC$ be a cocomplete closed symmetric monoidal category as in the previous subsection. We briefly review the basic definitions for $\CC$-valued algebraic quantum field theories on $\Loc$ following [@BSWoperad]. Algebraic quantum field theories with $\ast$-involutions are defined later in Section \[subsec:Involutions\]. We also refer to [@Brunetti; @FewsterVerch; @AQFTbook] for a broader introduction to algebraic quantum field theories and their applications to physics.

Let us denote by $\Alg:=\Alg_{\As}(\CC)$ the category of associative and unital algebras in $\CC$. An [*algebraic quantum field theory*]{} $\AAA$ on $\Loc$ 