The uni-molecular chemical reaction is here represented by the natural transformations $\eta:h^{A}\longrightarrow h^{B}$, through the following commutative diagram:

with the states of the molecular sets $Au=a_{1},\ldots,a_{n}$ and $Bu=b_{1},\ldots b_{n}$ being represented by certain endomorphisms in H(A,A) and H(B,B), respectively.

The observable of an $m.c.v$, $B$, characterizing the products “$B$” of a chemical reaction is defined as a morphism:

where R is the set of real numbers. This mcv-observable is subject to the following commutativity conditions:

with $c:A^{*}_{u}\longrightarrow B^{*}_{u}$, and $A^{*}_{u}$, $B^{*}_{u}$ being specially prepared fields of states, within a measurement uncertainty range, $\Delta$.

Note that in the case of either uni-molecular or multi-molecular, reversible reactions one obtains a quantum-molecular groupoid, QG, defined as above in terms of the mcv-observables. In the case of an enzyme, E, with an activated complex, $(ES)^{*}$, a quantum biomolecuar groupoid can be uniquely defined in terms of mcv-observables for the enzyme, its activated complex $(ES)^{*}$ and the substrate, S. Quantum tunnelling in $(ES)^{*}$ then leads to the separation of the reaction product and the enzyme, E, which enters then a new reaction cycle with another substrate molecule S’, indistinguishable–or equivalent to–S. By considering a sequence of two such reactions coupled together,

$QG_{1}\leftrightarrows QG_{2}$,

corresponding to an enzyme f, coupled to a ribozyme $\phi$, one obtains a quantum-molecular realization of the simplest (M,R)-system, $(f,\phi)$

The non-reductionist caveat here is that the relational systems considered above are open ones, exchanging both energy and mass with the system’s environment in a manner which is dependent on time, for example in cycles, as the system ‘divides’–reproducing itself; therefore, even though generalized quantum-molecular observables can be defined as specified above, neither a stationary nor a dynamic Schrödinger equation holds for such examples of ‘super-complex’ systems. Furthermore, instead of just energetic constraints–such as the standard quantum Hamiltonian–one has the constraints imposed by the diagram commutativity related to the mcv-observables, canonical functors and natural transformations, as well as to the concentration gradients, diffusion processes, chemical potentials/activities (molecular Gibbs free energies), enzyme kinetics, and so on. Both the canonical functors and the natural transformations defined above for uni- or multi- molecular reactions represent the relational increase in complexity of the emerging, super-complex dynamic system, such as, for example, the simplest (M,R)-system, $(f,\phi)$.