The distinct role of the Cu(I) in the Huisgen dipolar cycloaddition of azides to alkynes (denoted as CuAAC) is disclosed by following the evolution of the topology of the Laplacian of the electronic charge density, del(2)rho(r), and its gradient vector field, del del(2)rho(r), along the reaction coordinate with several density functionals (wB97XD, LCwPBE, M06-2X, M06-L, B3LYP) and the 6-311++G(d,p) basis set. Remarkably, in view of the topology of del(2)rho(r) and del del(2)rho(r), the mechanism appears to be diverse (asynchronous concerted or stepwise) depending on the reaction conditions. Overall, the catalyst orchestrates first the formation of the external N-C and subsequently the internal one by following alternatively a pericyclic-like or a pseudopericyclic-like mechanism. The role of the catalyst is envisaged as transforming the type of the mechanism from pericyclic to pseudopericyclic, and thence eventually facilitating the process. The mononuclear process (CuAAC) is concerted (with L = CH3CN) with all the functionals tested (i.e., wB97XD, LCwPBE, M06-2X, M06-L), except for the B3LYP who rendered a stepwise mechanism. Nevertheless, with L = H2O and CH3OH attached to the copper, the process becomes asynchronous concerted. Interestingly, upon introduction of the second Cu (Cu(2)AAC) at our best theory level (i.e., LCwPBE/6-311++G(d,p)), all the processes considered turned out to be concerted except for the 1,4-Cu(2)AAC which is predicted to be stepwise, with an extremely low enthalpy for the ring-contraction process (0.18 kcal/mol). This fact is explicated by the stability of the intermediate, which is in turn rationalized by the hole within the valence shell of the carbon attached to the Cu and the position of the (3,-3)(LP), of the internal N, toward it. Furthermore, due to the tiny energy difference between the stepwise dinuclear and concerted mononuclear mechanisms (0.39 kcal/mol), we argue that the concurrence of both processes (CuAAC and Cu(2)AAC) is feasible.