In order to investigate the nature of the insulator phase of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{Cu}}_2{\left(\mathrm{C}\mathrm{N}\right)}_3$ [where BEDT-TTF represents bis(ethylenedithio)terathiofulvalene] and why this salt is a superconductor under pressure but not under ambient pressure, we examined the ${}^{13}\mathrm{C}$ NMR spectra and spin lattice relaxation rate of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{Cu}}_2{\left(\mathrm{C}\mathrm{N}\right)}_3$ under pressure. The spin susceptibility evaluated from the Knight shift decreased with increase in pressure, and line broadening was suppressed above a critical pressure. The temperature dependence of ${\left(T_{1}T\right)}^{-1}$ for $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{Cu}}_2{\left(\mathrm{C}\mathrm{N}\right)}_3$ above a critical pressure was that for $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_{2}X$, where $X$ was $\mathrm{Cu}\left[\mathrm{N}{\left(\mathrm{C}\mathrm{N}\right)}_2\right]\mathrm{Br}$ or $\mathrm{Cu}{\left(\mathrm{N}\mathrm{C}\mathrm{S}\right)}_2$. These results suggest that the electronic structure of $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{Cu}}_2{\left(\mathrm{C}\mathrm{N}\right)}_3$ under pressure is similar to that of other $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_{2}X$ salts. The results support our model that electron inhomogeneity and hopping conduction, which are common features of semiconductivity in $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_{2}X$ at high temperatures, are responsible for the insulator phase in $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{Cu}}_2{\left(\mathrm{C}\mathrm{N}\right)}_3$. The electron inhomogeneity suppresses the semiconductor-metal transformation and superconductivity observed in other $\kappa \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_{2}X$ salts.