details.)

Additionally, we also assume that we have, for each object $\boldsymbol{M}$ of [Loc]{}, a quantum field $\Phi_{\boldsymbol{M}}$, thought of as describing some physics happening in $\boldsymbol{M}$. More precisely, we assume that there is a topological $*$-algebra $\mathscr{A}(\boldsymbol{M})$ (with unit) and that $f \mapsto \Phi_{\boldsymbol{M}}(f)$, $f \in C_0^\infty(\boldsymbol{M})$, is an operator-valued distribution taking values in $\mathscr{A}(\boldsymbol{M})$. (This would correspond to a “quantized scalar field” which we treat here for simplicity, but everything can be generalized to more general tensor- and spinor-type fields, as e.g. in [@VerSPST].) One would usually require that the $\Phi_{\boldsymbol{M}}(f)$ generate $\mathscr{A}(\boldsymbol{M})$ in a suitable sense, if necessary allowing suitable completions. The unital topological $*$-algebras also form a category which shall be denoted by [Alg]{}, where the morphisms $\mathscr{A} \overset{\alpha}{\longrightarrow} \mathscr{B}$ are injective, unital, topological $*$-algebraic morphisms between the objects. Then one says that family the $(\Phi_{\boldsymbol{M}})$, as $\boldsymbol{M}$ ranges over the objects of [Loc]{}, is a [*local covariant quantum field theory*]{} if the assignments $\boldsymbol{M} \to \mathscr{A}(\boldsymbol{M})$ and $\mathscr{A}(\psi)(\Phi_{\boldsymbol{M}}(f)) = \Phi_{\boldsymbol{N}}(\psi_*f)$, for any morphism $\boldsymbol{M} \overset{\psi}{\longrightarrow} \boldsymbol{N}$ of [Loc]{}, induce a functor $\mathscr{A} : {\sf Loc} \to {\sf Alg}$. (See again Fewster’s contribution [@cjfgm] for a fuller dish.) Actually, a generally covariant quantum field theory should 