How to use first_value() window function to get last non null value?

Question

I want to perform a fill-down operation in Postgres.

DDL:

create table brands (
id int,
category varchar(20),
brand_name varchar(20)
);

insert into brands values
 (1,'chocolates','5-star')
,(2,null,'dairy milk')
,(3,null,'perk')
,(4,null,'eclair')
,(5,'Biscuits','britannia')
,(6,null,'good day')
,(7,null,'boost')
,(8,'shampoo','h&s')
,(9,null,'dove')
;

Expected output is:

category	brand_name
chocolates	5-star
chocolates	dairy milk
chocolates	perk
chocolates	eclair
Biscuits	britannia
Biscuits	good day
Biscuits	boost
Shampoo	h&s
Shampoo	dove

I tried using the following script but it doesn't seem to work.

select id,
      first_value(category)
      over(order by case when category is not null then id end desc nulls last) as category,
      brand_name
from brands

Can someone suggest a fix?
In MS SQL the following snippet seems to work fine:

select id,
       first_value (category) IGNORE NULLS
       over(order by id desc
        rows between current row and unbounded following) as category,
       brand_name
FROM brands
ORDER BY id

Answer 1

with cte as (
select id,
       category,
       count(category) over (order by id) as category_id,
       brand_name
  from brands)
select id,
       first_value(category) over (partition by category_id order by id) as category,
       brand_name
  from cte;

UPDATE: added query without CTE per request:

select id,
       (array_agg(category) over (order by id))[max(case when category is null then 0 else id end) over (order by id)] as category,
       brand_name
  from brands;