Add an incremental categorical column

I have a table with some production value and one of the column shows the type of product made. In my example below I am changing the type of product twice and I run the same product twice.

I would like to add a column that would help me compare the first run of prod-X with the second run of prod-X by adding a column named run and the value would be incremental run-1, run-2 and run-3.

How do I update this table to add those run index value to the row?

Thanks

Datetime            Width type                        Run
-----------------------------------------------------------
2024-01-01 06:00:00 1.45  prod-X --Change of product  run-1
2024-01-01 06:10:00 1.47  prod-X                      run-1
2024-01-01 08:00:00 2.56  prod-Y --Change of product  run-2
2024-01-01 08:10:00 2.67  prod-Y                      run-2
2024-01-02 07:00:00 1.87  prod-X --Change of product  run-3
2024-01-02 07:10:00 1.94  prod-X                      run-3

This is a classic gaps-and-islands problem where:

• you first find the change using LAG
• you then find the group by using SUM over the number of changes so far e.g.

create table MyTable([DateTime] datetime2(0), Width decimal(9,2), Type varchar(12));

insert into MyTable([DateTime], Width, Type)
values
('2024-01-01 06:00:00', 1.45, 'prod-X'),
('2024-01-01 06:10:00', 1.47, 'prod-X'),
('2024-01-01 08:00:00', 2.56, 'prod-Y'),
('2024-01-01 08:10:00', 2.67, 'prod-Y'),
('2024-01-02 07:00:00', 1.87, 'prod-X'),
('2024-01-02 07:10:00', 1.94, 'prod-X');

with cte1 as (
  select *
    -- Detect a change in type when the previous row is different
    , case when type <> lag(type, 1, type) over (order by DateTime asc) then 1 else 0 end gp
  from MyTable
), cte2 as (
  select *
    -- Keep a running sum of the change in type to get unique groups
    , sum(gp) over (order by DateTime asc) + 1 groupnum
  from cte1
)
select [DateTime], Width, Type
  -- Build the required display label
  , 'run-' + convert(varchar(3), groupnum)
from cte2
order by [DateTime] asc;

db<>fiddle