A person who insists on something can be described as:
- Insistent
- Persistent
- Determined
- Adamant
- Stubborn (depending on context, sometimes with a negative connotation)
- Tenacious
If you want a noun to describe such a person, you might say:
- Stickler (someone who insists on a certain quality or behavior)
- Perfectionist (if they insist on things being perfect)
- Advocate (if they insist on a cause or idea)
- Zealot (if they insist with extreme passion, sometimes fanatically)
Would you like examples or synonyms in a particular context?
