Step 1: Reading the Question:
We must name the persons under whose supervision Hamza Nama was completed.
Step 2: Method:
Sort the options into rulers and painters. A project is supervised by rulers, while painters do the work. Then match the rulers with the dates of the manuscript.
Step 3: Going through the choices:
Choice 1: Humayun and Akbar are both rulers, and the manuscript started in Humayun's time and ended in Akbar's. Accepted.
Choice 2: Babur and Humayun are rulers, but Babur's reign ended before the project began. Rejected.
Choice 3: Ustad Mansur was a painter of Jahangir's time, so he cannot be the supervising authority. Rejected.
Choice 4: Daswant was a painter, not a ruler who supervised. Rejected.
Step 4: Conclusion:
The ruler pair that fits the timeline is Humayun and Akbar.
\[ \boxed{\text{Option 1: Humayun and Akbar}} \]