The designer dressed them in black and teal costumes. Determine the case and function of the underlined pronoun

Even though you didn't underline anything, there is only one pronoun in this sentence - the word them.

The case of this pronoun is objective - it functions as an object in the sentence. It is really easy to determine this - all you have to do is ask the questions whom or what. Whom did the designer dress in black and teal costumes? And the answer is - Them.

This way you know that the pronoun is also the object; to be more specific, it is a direct object.

Therefore, the correct answers are case: objective; function: direct object.