First, recall that a monopolist raises prices in order to make higher profits than a competitive firm would in the same market. They are able to do so because no competition exists to undercut their prices.
If a competitor emerges, they may undercut the monopoly's price, thereby drawing customers away and forcing the monopoly firm to reduce their prices. That means less profit for the monopolist--so obviously they will not be happy about this.
However, there are some options the monopoly firm has in order to avoid this outcome. With only one other firm competing with them, they are in duopoly, which is almost a monopoly and can be made to behave like one. The two firms may attempt to collude, agreeing (either explicitly, which is usually illegal, or tacitly, which is legal but harder to sustain) to maintain the higher monopoly price and split the profits.
There is an incentive for each firm to break this collusion by undercutting the other company's price; but if they do, they know that they will create a price war that eventually drives prices down to the competitive level. If they expect the other firm to stick around for a long time, they can actually enter an equilibrium where both firms tacitly collude in order to avoid being punished by the other firm later.
If collusion is not an option, the monopoly might instead try regulatory capture, lobbying the government to make regulations in their favor, perhaps complex licensing requirements that only an established firm is capable of meeting. In effect they use the government to shut down the other company and restore their monopoly.