Update Nested JSON Components in Postgres

I have a table that looks like this:

parts
id     name       children
1      Part 1     [{id: 2, name: 'Part 2'}]
2      Part 2     []

Let's say I change the name of Part 2 to Part X

I want to update both the main data, and the nested children data.

I did this:

UPDATE parts SET name = 'Part X' WHERE id=2

Now how do I also update the JSON data to match?

I thought this was good syntax, but it's not:

UPDATE parts SET [{name: 'Part X'}] WHERE json_agg([id: 2])

Also this is Postgres v10