To identify a redox reaction, we must check if there is a transfer of electrons between substances (oxidation and reduction).
Option A: \( \mathrm{H_2CO_3 \rightleftharpoons H^+ + HCO_3^-} \)
This is not a redox reaction. It is an equilibrium dissociation of carbonic acid, where no electron transfer occurs.
Option B: \( \mathrm{Hg^{2+} + 2OH^- \rightleftharpoons Hg(OH)_2} \)
This involves the precipitation of mercury (II) hydroxide but does not involve any electron transfer. Hence, it is not a redox reaction.
Option C: \( \mathrm{C_6H_{12}O_6 + 6O_2 \rightleftharpoons 6CO_2 + 6H_2O} \)
This is a classic example of a redox reaction, known as cellular respiration. In this reaction, glucose is oxidized to carbon dioxide and oxygen is reduced to water. Thus, electrons are transferred, making it a redox reaction.
Option D: \( \mathrm{CaCO_3(s) \rightleftharpoons Ca^{2+} + CO_3^{2-}} \)
This is a dissociation reaction, not a redox reaction. There is no electron transfer here.
Thus, the correct redox reaction is Option (C).