Search for products in all categories via ajax search form

[ikran]

Good morning,
on my ecommerce site I have many products (about 250,000) divided into categories as in the example below:

MAIN Category
            - subcategory 001
                         - subcategory A
                         
in the MAIN category I left only some products, others only in subcategory 001 and others only in subcategory A ...
this is because in this way the loading of the pages when opening any category is faster.

On the site I also have an ajax search form that is displayed both on the homepage and when you enter a specific category.

When I search from the home page, the products are always found (without influence of the categories), while if I open the MAIN category and perform a search I find only the products of the MAIN category and not those of the other categories.

Is it possible to change the search so that products are searched for in all categories?