Efficiently converting carbon dioxide (CO$_2$) into valuable fuels or chemicals represents one of the great challenges at the core of current scientific researches. Here, we report a DFT-based theoretical study of the reactivity of two main MoS2024-06-10 edges for electrocatalytic, or eventually photocatalytic, reduction of CO2024-06-10. By explicitly accounting for the electrode potential via a grand-canonical ensemble that controls the number of electrons, we show that the two edges exhibit different H coverage which, in turn, directly influences the CO2024-06-10 -reduction energy profile under relevant reducing potentials. Specifically, on S-edge, a 0.375 ML H coverage enables the CO$_2$ activation through its adsorption in a bidentate mode with a limiting potential of 0.07 V vs. SHE. By contrast, H coverage on Mo-edge at reducing potentials relevant for CO2024-06-10 reduction was determined to be 0 ML. On this bare Mo-edge, CO$_2$ activation occurs at an additional energy cost of 0.44 eV for -0.80 V. Consequently, the activated CO$_2$ on S-edge exhibits comparable and thermodynamically favorable reactivity for the two possible two-electron products, formic acid and carbon monoxide (CO), with limiting potentials of -0.54 V and -0.34 V, respectively. The two products, however, show different desorption behavior with CO desorption being endergonic and independent of electrode potential. Conversely, the reactivity of Mo-edge is less favored and it is anticipated to favor formic acid over CO due to a highly endergonic C-O bond-breaking step in the adsorbed COOH intermediate. This step was determined to be exergonic on S-edge.This study paves the way for a better understanding of CO$_2$ reduction mechanisms on MoS$_2$ edges and highlights the key role of the CO$_2$ adsorption step which was generally neglected in prior investigations.