I think (I'm not certain) that foreign key constraints won't do precisely what you want given your table design. Perhaps the best thing to do is to define a stored procedure that will delete a category the way you want, and then call that procedure whenever you want to delete a category.
CREATE PROCEDURE `DeleteCategory` (IN category_ID INT)
LANGUAGE SQL
NOT DETERMINISTIC
MODIFIES SQL DATA
SQL SECURITY DEFINER
BEGIN
DELETE FROM
`products`
WHERE
`id` IN (
SELECT `products_id`
FROM `categories_products`
WHERE `categories_id` = category_ID
)
;
DELETE FROM `categories`
WHERE `id` = category_ID;
END
You also need to add the following foreign key constraints to the linking table:
ALTER TABLE `categories_products` ADD
CONSTRAINT `Constr_categoriesproducts_categories_fk`
FOREIGN KEY `categories_fk` (`categories_id`) REFERENCES `categories` (`id`)
ON DELETE CASCADE ON UPDATE CASCADE,
CONSTRAINT `Constr_categoriesproducts_products_fk`
FOREIGN KEY `products_fk` (`products_id`) REFERENCES `products` (`id`)
ON DELETE CASCADE ON UPDATE CASCADE
The CONSTRAINT clause can, of course, also appear in the CREATE TABLE statement.
Having created these schema objects, you can delete a category and get the behaviour you want by issuing CALL DeleteCategory(category_ID) (where category_ID is the category to be deleted), and it will behave how you want. But don't issue a normal DELETE FROM query, unless you want more standard behaviour (i.e. delete from the linking table only, and leave the products table alone).