The mechanism of a bioortogonal click reaction between 2-azidoethanol and coumBARAC has been examined using DFT methods at B3LYP/6-31G(d,p) and M06-2X/6-31G(d,p) computational levels. The regiochemistry of the reaction has been studied based on potential energy surface analysis and global reactivity indices of the reactants. The global electron density transfer (GEDT) calculations at the possible transition states (TSs) revealed that this cycloaddition (CA) has a nearly non-polar character. The strain and electronic effects on the reactivity of coumBARAC with 2- azidoethanol was studied using distortion/interaction transition state model and compared with CA reaction of this dipole with phenylacetylene as a strain-free reaction. The IRC calculations and the topological electron localization function (ELF) analysis explain properly the one-step two-stage mechanism of this strain-promoted CA reaction, as well as the effects of the strain on the electronic structure of cycloalkynes and the activation energy barriers of CA reactions. By application of bonding evolution theory (BET), nine successive domains of structural stability have been recognized along the reaction path, as well as the bifurcation catastrophes responsible for the changes in the topology of the system.