There exist a variety of multi-pulse NMR experiments for spectral assignment of complex molecules in solution. The conventional heteronuclear multiple-quantum coherence (HMQC) NMR experiment provides correlation between weakly coupled hetero-nuclei. The COSY is one of the most Popular two-dimensional NMR experiment which is used to correlate J-coupled homo-nuclei of spectral assignment. The combination of the conventional HMQC and COSY NMR experiments yields a new experiment called 3D HMQC-COSY NMR experiment. The product operator theory is widely used for the analytical descriptions of multi-pulse NMR experiments for weakly Coupled spin systems in liquids. In this study, complete product operator theory for weakly coupled IS (I = 1, S = 1) spin system is presented by obtaining the evolutions of the product operators under the spin-spin coupling Hamiltonian. As an application and a verification, analytical descriptions of 3D HMQC-COSY NMR experiment are obtained for weakly coupled ISnI'S-m' (I = I' = 1/2; S = S' = 1: n = 1,2,3; m = 1,2) multi-spin systems. Then the estimated spectra of this experiment for various multi-spin systems are explained in detail.