1. Reaction 1: Ca(OH)$_2$ reacts with NH$_3$Cl to form NH$_3$, CaCl$_2$, and H$_2$O.
2. Reaction 2: NH$_3$ reacts with H$_2$O and CO$_2$ to form NH$_4$HCO$_3$.
Thus, A = Ca(OH)$_2$, B = NH$_3$, and C = NH$_4$HCO$_3$.
Ammonium bicarbonate (NH$_4$HCO$_3$) is formed when ammonia reacts with water and carbon dioxide, a key step in the Solvay process.