How do I change the markup only of a built-in widget?

I’m developing a theme and I’d like to slightly alter some of the HTML output of some of the built-in widgets. As of now, I managed to change the sidebar search widget by creating a searchform.php file in my theme — though that might not only be the sidebar widget but alright.

I’m now onto the Archives widget. I literally just need to alter the bit that says “Archives” to change its tag and add a class. Do I really need to make my own (largely identical) class and register it and use my custom widget instead?

(If you can’t tell, I haven’t touched WP theming in around 10 years 🙂

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Author: Morpheu5