To add to f6pafd's correct answer, "a cat" is object to the preposition "to" within the phrase "to a cat" that complementizes the head of the subject complement "equal" in the sentence A dog is equal to a cat. "A cat" would be part of the subject if placed before the verb:
- A dog and a cat are equal.
And it would be an object if placed after a transitive verb.
- A dog chases a cat.
When a copulative verb such as be is used, the subject and the subject complement are identified with the same referent (I = a cat), but syntactically speaking only what comes before the verb is the subject—what follows is, as f6pafd said, the subject complement.