Somehow, how can categorize it the by A ,B and C hospital
The easiest way is to create a centralized view, which returns the data for all hospitals incl. a sign, from which hospital the data comes from, like
SELECT 'A' AS HospitalName, FirstName, LastName, Room
FROM HospitalDB_A.dbo.Patient
UNION ALL
SELECT 'B' AS HospitalName, FirstName, LastName, Room
FROM HospitalDB_B.dbo.Patient
UNION ALL
SELECT 'C' AS HospitalName, FirstName, LastName, Room
FROM HospitalDB_C.dbo.Patient
If you get a new hospital, you only need to modify the view.