Reconciling a column across two tables in SQL Server

UPDATE Table_B
SET Table_B.Flag = Table_A.Flag
FROM Table_A inner join Table_B on Table_A.id = Table_B.id

UPDATE
  DatabaseB.dbo.Table_B
SET
  DatabaseB.dbo.Table_B.[Flag] = DatabaseA.dbo.Table_A.Flag
FROM
  DatabaseA.dbo.Table_A inner join DatabaseB.dbo.Table_B B
  on DatabaseA.dbo.id = DatabaseB.dbo.B.id

create trigger update_b_trigger
on Table_A
for update
as
begin
   update Table_B
   set Table_B.flag = inserted.flag
   from inserted
          inner join Table_B
             on inserted.id = Table_B.id
                 and inserted.group = 'B'
                 and inserted.flag <> Table_B.flag
end