This hospital is known for having a lot of patience. Is this a homographic pun or homographic pun or both

This statement is a homographic pun. The phrase "a lot of patience" is a play on words, as it can be interpreted to mean both "many patients" in the context of a hospital, and also "having patience" as a virtue. Homographic puns rely on words that have the same spelling but different meanings to create a humorous or clever effect.