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!
Given any enumeration $E$ of the language, here is a way to describe a language: "the 5th language in the enumeration".

If you use that as the description, then yes, your condition is trivially true.  Given the description of a language ("the 5th language in $E$"), you can quickly find its position in $E$ (5).