Many people say they hate political correctness. Of course, moral libertarians naturally hate political correctness too. The principle of equal moral agency requires that everyone be able to express and promote their moral views, which means that free speech is needed. Furthermore, we believe in the free market of ideas, which requires a lack of impediment to free speech to function properly.
Among people who say they hate political correctness, however, many fail to oppose all forms of political correctness, or worse, fail to even recognize all forms of political correctness. For example, in last year's Australian Marriage Law Postal Survey, the 'no' camp said that extending marriage rights to same-sex couples was political correctness. But was that really true? After all, same-sex couples already lived in committed relationships recognised by society, and often had legally unrecognised wedding ceremonies. Same-sex couples already lived in marriage-like relationships, and their relationships were generally regarded across Australian society as not different in nature from heterosexual marriages, as evidenced by the landslide victory of the 'yes' camp. Hence the idea of marriage being a relationship between two people regardless of gender was already de-facto correct, it just wasn't legally correct, i.e. politically correct. Amending the law would bring legal and political correctness in line with the reality. Conversely, the idea that marriage could only be between a man and a woman had already become only politically correct, because it did not line up with lived reality in Australian society anymore. Hence, the 'no' camp was the politically correct camp, hoping to maintain a standard of political correctness that deviated from reality. As you can see, conservatives are not free from political correctness either, they are just blind to it: so blind that they often accuse the other camp of being politically correct instead.
In fact, the left is also responsible for some of the right's attitude that political correctness is whatever they don't like. A substantial part of the left have bought the idea that political correctness is who they are, so much that when former Democratic presidential candidate Bernie Sanders blamed Donald Trump's victory on political correctness, the comment were controversial within the left. In this case, the political correctness Sanders was referring to surrounded the lack of criticism of pro-corporate policy in mainstream media and politics, something both socialists and libertarians alike have been complaining about for quite a long time. Therefore, there was no reason why the left would disagree with him here. In fact, the left didn't disagree with Sanders at all, they were just not used to describing their opponents as politically correct. But Sanders was clearly correct here. Mainstream politics had presumed a 'correct' point of view and failed to represent other points of view, and this is political correctness by definition.
Having discussed two examples of right-wing political correctness, I believe I need to provide some balance here. Of course, the left is not without problems of political correctness either, to put it mildly. In recent years, the left has indeed taken leftist political correctness to new heights, with concepts like no-platforming and safe speech. Politically incorrect speech is now deemed unsafe, and must be no-platformed, i.e completely disallowed. This attitude, formerly believed only to exist in fascist dictatorships, is invading progressive circles at a worrying pace. Reviving the free speech culture and upholding the free market of ideas has arguably become our most urgent imperative, as moral libertarians. As a former US president liked to say, freedom is never more than two generations from extinction.
In conclusion, political correctness comes from both the left and the right, and perhaps even other directions too. To oppose political correctness sincerely is to oppose all forms of political correctness consistently. We cannot allow being against political correctness to become just a brand politicians use to attack opponents, or a slogan the right use to attack the left.