Enumeration of a class of languages

Can you enumerate a class of languages in such a way that the description of every language/ machine enumerated encodes where it was in the enumeration?

Ex:if you are given the description of the bth machine/language enumerated you could quickly check, by some tag or marker in the description of the bth machine / language, that it was indeed the bth machine/language in the enumeration by an enumerator E.

Can you modify any enumerator for a class of languages so that it does this? Or can such an enumerator E be rigged up to do this for any enumerable class of languages?

To give context, my goal is for a verifier to accept, as one part of its input, the machine/language enumerated, and be able to see, quickly, what position that machine/language was in the enumeration by the enumerator E.

Thank you!