A new porous metal-organic framework, [Co (oba) (bpdh)]·(DMF) (TMU-63), containing accessible nitrogen-rich diazahexadiene groups was successfully prepared with the solvothermal assembly of 5-bis(4-pyridyl)-3,4-diaza-2,4-hexadiene (4-bpdh), 4,4'-oxybis(benzoic) acid (oba), and Co(II) ions. The combination of Lewis basic functional groups and porosity leads to high performance in CO2 adsorption and conversion in the cycloaddition reaction of epoxides under solvent-free conditions. To further enhance the catalytic efficiency of TMU-63, we introduced a highly acidic malonamide ligand into the structure via solvent-assisted ligand exchange (SALE) as a postsynthesis method. Incorporating different percentages of N1,N3-di(pyridine-4-yl) malonamide linker (4-dpm) into TMU-63 created a new porous structure. Powder X-ray diffraction (PXRD) and NMR spectroscopy confirmed that 4-bpdh was successfully replaced with 4-dpm in the daughter MOF, TMU-63S. The catalytic activity of both MOFs was confirmed by significant amounts of CO2 cycloaddition of epoxides under solvent-free conditions. The catalytic cycloaddition activities were found to be well-correlated with the Lewis base/Brønsted acid distributions of the materials examined in the TMU-63S series, showing that the concurrent presence of both acid and base sites was desirable for high catalytic activity. Furthermore, the heterogeneous catalysts could easily be separated out from the reaction mixtures and reused four times without loss of catalytic activity and with no structural deterioration.