One-pot three-component regioselective azide-alkyne cycloadditions are central reactions for synthesizing pharmaceuticals and fine chemicals and are also applied for in vivo metabolic labeling biotechnology. Homogeneous catalysts based on copper species coordinated with ancillary ligands are regularly used to perform this reaction, offering superior catalytic activity and selectivity compared to conventional heterogeneous counterparts based on supported copper nanoparticles. However, the challenge of catalyst recovery limits the use of these homogeneous compounds in many large-scale applications. In this work, we report the high catalytic performance of a family of Cu-based single-atom catalysts for triazole synthesis, with an emphasis on the fundamental understanding of the structure and function of the catalyst. The catalysts were prepared via tricyanomethanide polymerization to create a joint electronic structure where the mesoporous graphitic carbon nitride carrier acts as a ligand for the atomically dispersed copper species. The material properties and the precise metal location/coordination (i.e., deposited in the heptazine pore of carbon nitride, substituted in the framework of carbon nitride, hosted in a vacancy, or entrapped in sandwich-like arrangement) were characterized through a battery of spectroscopic and theoretical methods. The catalysts were employed in the synthesis of 1,2,3-triazoles employing azide-alkyne click reaction under base-free conditions. The single-atom Cu catalysts demonstrated improved activity and selectivity compared to the homogeneous reference catalyst. Density functional theory calculations corroborated the results and showed that the reaction proceeds through a barrier given by the activation of the acetylenic moiety on Cu₁. The activity of this step was primarily affected by the coordination of the metal with the support. Therefore, understanding the metal coordination in single-atom catalysts is critical to further optimizing single-atom catalysts and greening synthetic chemistry.