First, WordPress will look for a template specific for that particular category using the category slug. For example, category-design.php template will be used to display the ‘Design’ category.
If it does not find a category-slug template, then WordPress will look for a template with category id, for example, category-6.php. After that, it will look for the generic category template which is usually category.php.
If there is no generic category template present, then WordPress will look for a generic archive template, such as archive.php. Lastly, it will use index.php template to display the category.