Isomerization dynamics involving the migrations, proton transfer reaction, and catalytic actions of water molecules upon vertical ionization of the formamide (FA)-(H₂O)₂ cluster is investigated by the infrared spectroscopy and theoretical reaction path search calculation. The infrared spectroscopic result indicates the [FA-(H₂O)₂]⁺ cation has the hydrogen-bonded structure of the enol isomer cation of formamide and the water dimer. This structure is formed by proton transfer from the CH bond to the carbonyl group through the catalytic action of the water molecules. The isomerization paths involving this enolization in ionized FA-(H₂O)₂ are explored by using the anharmonic downward distortion following method. We found multiple enolization paths which accompany proton exchanges among the formamide moiety and water molecules through the catalytic actions of the water molecules.