First let us rephrase the question:

Whose vicegerent was Prophet Adam and whose vicegerent is man? And keeping in mind that one who is vicegerent is responsible for all the duties and tasks of whom he is taking the place of, it is necessary for him to bear his same qualities. How much potential has Allah (swt) granted man that will allow him to bear similar qualities to Allah’s (swt) and be responsible for some of His responsibilities? How much can man be a reflection of Allah (swt)?

Whose vicegerent is man[1]?

Considering verses 30 and 31 of surah Baqarah, man is the vicegerent of Allah (swt).

Explanation: In verse 30, the phrase “انی جاعل فی الارض خلیفة” and the single pronoun in it all show that Allah (swt) wants to appoint a vicegerent for Himself on earth, not for anyone else, and in the next verse[2], He discloses that the reason for appointing man to this important rank is because of his knowledge regarding the ‘divine names’ and says: “و علّم آدم الاسماء کلها” and in other words, in the first verse, Allah (swt) announces the appointment of man as vicegerent and in the second, unveils man’s jurisdiction and the scope of his vicegerency and since Allah's (swt) essence manifests in His names, through explaining what these names are and what it means that Allah (swt) taught man these names, it will become clear who exactly man is the vicegerent and how much he can do godly things and bear godly qualities.

The meaning of ‘divine names’

In Arabic, ‘name’ or ism means ‘sign’. Words (verbal names) are signs of concepts and meanings, and these meanings are signs of both seen and unseen realities that are signs of their creator. Now what needs to be determined is which of these categories were the signs man was taught; words, concepts and ideas, or realities?

Knowing mere words and names, or concepts aren't enough to make the angels prostate towards man, because “The most words can do is help us understand what others have inside and want to share, while angels aren't in any need of words, because they understand what others have on the inside without any need of discourse.”[3]

The mere understanding of concepts and having the form of realities and entities in the mind is also something that angels are higher than, because they are connected with some of the realities of this world instead of only having their form with them like we do, and in other words, they have intuitive knowledge (Ilm Hudhuri) of these realities instead of ‘acquired knowledge’ (Ilm Husuli), although this isn't the case for them regarding all realities. Thus, the reason for their prostration can't just be because of man’s knowledge regarding the forms of realities. So, what is meant by the names that man was aware of are those unseen realities and all other concrete and seen realities in which the angels couldn’t bear altogether and don’t have the capacity of reaching the pinnacle of and according to the ensuing verses, those names are the unseen realities of this world, because in verse 33, Allah (swt) says: “الم اقل لکم انی اعلم غیب السموات و الارض”,[4]; according to the interpretation of some tafsirs, what the context of the three verses of 30-33 calls for is that the “غیب” mentioned in verse 33 that Allah (swt) says he knows, is none other than the realities taught and spoken of in the phrase “انی اعلم ما لا تعلمون”[5], meaning that Allah (swt) has given man the knowledge of all the things that the angels are unaware of and only He knows of. So the ‘names’ and ‘signs’ are the unseen realities that transcend the comprehension of the angels who themselves are above and higher than the material world and bear some degrees of the unseen world; these realities are the same high realities that other realities originate from and are covered by, let them be of the seen or unseen.[6]

The meaning of ‘teaching’

Considering what was said and Prophet Adam (pbuh) receiving the ‘signs’ directly,[7]one can conclude that what is meant by ‘teaching’ (that the verse has said) is:

1- Endowing him with the intuitive knowledge of these realities, not just the mental awareness of their forms and

concepts.

2- The knowledge being transmitted is divine, and the transmitter is Allah (swt) Himself, and since knowledge is a type of possession, both in acquired knowledge (where we have the possession of the form and idea of a reality)  and in intuitive knowledge (where we have possession of the reality itself instead of its idea), the two traits just mentioned for ‘divine teaching’ show that Allah (swt) provided Prophet Adam (pbuh) and man with the concrete reality of His qualities and (according to the previous definition we mentioned for Ism which was ‘sign’) made him a complete reflection and sign of His own names and qualities.

Keeping the whole of what explained above in mind and also the fact that the verse says that man was taught all of the names, and the name that encompasses all of the names of God is “Allah”, one can conclude that man is the vicegerent of Allah, not Rabb, because Rabb is one of the names of God and one of the manifestations of “Allah”. In reality, man is the complete manifestation of Allah and all of His qualities and attributes, and is in charge of doing all the things Allah does regarding all of creation other than Himself.

For further information, see:

1- Tafsir Al-Mizan, under verses 30-34 of surah Baqarah.

2- Javadi Amoli, Tafsir Tasnim, vol. 3, pp. 17-321.