Join - 3 tables - products categories

[Pierre de LESPINAY]

Hello

Simplifying my problem:
I have 3 tables.

user_master
----------
int    user_id
char   user_name
 
product_master
----------
int    product_id
char   product_name
int    product_category
 
user_product
----------
int    user_id
int    product_id

Of course the users can own multiple products
I need to ouput for each of them the number of products of each
category
For example:

User      | Chair    | Table    | Desk
----------+----------+----------+-----------
Dupont    |        2 |        3 |        1
Dupuy     |        1 |        0 |        3
Martin    |        0 |        1 |        2

I'm on this way:

SELECT
usr.user_name 'Nom',
count(up1.product_id) 'Chaise',
count(up2.product_id) 'Table',
count(up3.product_id) 'Bureau',
pr.product_name 'Produit'
FROM
user_master usr
NATURAL LEFT JOIN user_product up1
NATURAL LEFT JOIN user_product up2
NATURAL LEFT JOIN user_product up3
NATURAL LEFT JOIN product_master pr
WHERE
up1.product_category = 1 AND
up2.product_category = 2 AND
up3.product_category = 3
GROUP BY Nom
ORDER BY Nom

That doesn't seems to be complex, but this query gives me an empty
result.
In fact, I don't understand the principal of the JOIN but I mainly
don't understand his results.

Thx in advance.