Construction of a Natural Transformation from a Classical to a Quantum 0-Species

A natural transformation $\mathfrak{J}$ between functors valued in the category ${\mathfrak{C}\mathfrak{h}\mathfrak{d}\mathfrak{v}}_0$ is assembled. ${\mathfrak{C}\mathfrak{h}\mathfrak{d}\mathfrak{v}}_0$ is obtained by replacing both the categories $\mathrm{p}\mathrm{t}\mathrm{l}\mathrm{s}$ and $\mathrm{p}\mathrm{t}\mathrm{s}\mathrm{a}$ with the category of topological linear spaces in the defining properties of the category $\mathfrak{C}\mathfrak{h}\mathfrak{d}\mathfrak{v}$ introduced in one of our previous papers. By letting a $\mathfrak{d}\mathfrak{p}$-valued functor be (classical) quantum whenever every its value is a dynamical pattern whose set map takes values in the set of (commutative) noncommutative topological unital $\ast -$algebras, and letting a (classical) quantum $0$-species be a ${\mathfrak{C}\mathfrak{h}\mathfrak{d}\mathfrak{v}}_0$-valued functor factorizing through the canonical functor from $\mathfrak{d}\mathfrak{p}$ to ${\mathfrak{C}\mathfrak{h}\mathfrak{d}\mathfrak{v}}_0$ into a (classical) quantum $\mathfrak{d}\mathfrak{p}$-valued functor, we have that the domain and codomain of $\mathfrak{J}$ are a classical and a quantum $0$-species respectively.