Automated adding of one tag to all the posts in a category

I’m currently redesigning a site that has 700+ posts, divided into about 150 different categories. Some categories have 2 posts in them, other categories have 70.

Now for every category, I want to add a tag to all the posts in that category.

Does anybody know if there is a way to automatically add a specific tag to all the posts in a category?

Or is the only way going through all the posts individually and adding a tag to each one, one by one?

Any help much appreciated!