In the sentence الولدُ الذكيُّ محبوبٌ ("The smart boy is loved"), the word الذكيُّ functions as a نعت (na't), which is an adjective or qualifier that describes the noun before it. In Arabic grammar, a نعت must agree with its منعوت (the noun it describes) in four key aspects: definiteness (تعريف/تنكير), case (إعراب), gender (تذكير/تأنيث), and number (إفراد/تثنية/جمع).
Here, الذكيُّ agrees with الولدُ in all four respects: both are definite (marked with ال), both are in the nominative case (مرفوع), both are masculine, and both are singular. This full agreement confirms that الذكيُّ is a نعت modifying الولدُ, not a separate syntactic element.
The other options are incorrect. A حال (circumstantial expression) describes the state of the subject or object at the time of the action and is typically indefinite (نكرة), whereas الذكيُّ is definite. An اسم موصول (relative pronoun) would be a word like الذي or التي, used to introduce a relative clause. A مفعول به (direct object) receives the action of a transitive verb, but this is a nominal sentence (جملة اسمية) with no verb, so there is no مفعول به.