How do I rename a category in the menu bar only?

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Actually this is pretty simple and straightforward: when you set up your menu (under Appearance > Menu) you will find out you can add Categories there.

After adding them, click on the down-pointing arrow on the right of the navigation element, it will give you the possibility to change the Label, while keeping the Original visible to you, so you remember what it actually refers to.

In order to make the Menu you just created appear in your theme, you have to look at the wp_nav_menu function. You can take to approaches to display your menu:

  1. Either you register your menu location in your theme, and then fill up the menu in your dashboard. In this case you will use the $theme_location argument of the function.

  2. Or you take the opposite direction: create a menu in your dashboard and use it in your theme, using the $menu argument, which accepts id, slug or name.