How to store a list of options in Mongodb?

Question

MongoDB

I'm trying to decide how best to organize the storage of a list of product categories:

Дуб
Береза
Клен
и тд

Accordingly, a product may contain one or more categories.
I'm thinking of creating two separate collections (products and categories). And in the product document, have the field:

categories: [ObjectCategoryId, ObjectCategoryId, ObjectCategoryId]

where and refer to categories by key. But there is also a need for a count next to each category.

A product can have no more than 50 categories. Does it make sense to use an embedded strategy that will require all documents to be updated if the categories change?