Before doing any database changes, you should back up your database, and test any changes in a test system first. If you make a mistake you can revert to the backup.

First, verify you are updating the right things:

select p.product_id, concat(substring(e.name, 1, 1), lower(substring(e.name, 2)))
from products p
join entities e
on e.entity_id = p.product_id
order by e.name;

This statement will show a listing of the product ids and what their updated names will look like. Note that products with multiple words will have subsequent words lowercased.  E.g.

AMOXIL SYRUP FORTE 50MG/ML -> Amoxil syrup forte 50mg/ml

If you don't want this, you'll probably have to write a stored procedure to do it.

select count(p.product_id)
from products p
join entities e
on e.entity_id = p.product_id;

This statement will count the maximum number of rows that will be updated. It is an upper bound - your actual update may change fewer rows as they might already have the correct case.

In my database, it shows 890.

begin;
update products p
    join entities e
    on e.entity_id = p.product_id
set e.name = concat(substring(e.name, 1, 1), lower(substring(e.name, 2)));

These two statements perform the update in a transaction. You will see something like:

841 row(s) affected Rows matched: 890  Changed: 841  Warnings: 0    0.016 s

In the above, 841 of the 890 rows were changed, because 49 already had the correct case.

Before you commit the changes, you can run:

select p.product_id, e.name
from products p
join entities e
on e.entity_id = p.product_id
order by e.name;

to verify that the products were updated correctly.

If you aren't happy with the results, run:

rollback;

to discard the update.

Otherwise run:

commit;

to make them permanent.