If you put solid Ca(OH)2 into water, both processes run at once. Eventually, a balance is struck between the two, and solid forms at exactly the same rate that it dissolves. There will be no apparent change in the amount of solid or in the concentrations of the calcium and hydroxide ions in the tank at that point. Equilibrium has been established.
Dumping more hydroxide ions into the solution will upset that equilibrium. More hydroxide ions mean more encounters between between calcium and hydroxide ions in solution, so more solid calcium hydroxide will form in the first process. The extra hydroxide won't directly affect the second process; it runs at the same rate as before. The net result is that some of the calcium hydroxide precipitates.
