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For fear that this version of the question would be downvoted and closed (for being imprecise and/or lacking details) I have tried to give such a list of properties down below. Anyone who has an answer to the main question (or even anyone at all) is free to read or not read the further content below. There are several questions formulated in the body of the text below but the main question and most important for me is still the one above. I emphasize that anyone who wishes to answer this question alone is free to ignore the content below in his answer.

For fear that this version of the question would be downvoted and closed (for being imprecise and/or lacking details) I have tried to give such a list of properties down below. Anyone who has an answer to the main question (or even anyone at all) is free to read or not read the further content below. There are several questions formulated in the body of the text below but the main question and most important for me is still the one above. I emphasize that anyone who wishes to answer this question alone is free to ignore the content below in his answer.

For fear that this version of the question would be shut down I have tried to give such a list of properties down below. Anyone who has an answer to the main question (or even anyone at all) is free to read or not read the further content below. There are several questions formulated in the body of the text below but the main question and most important for me is still the one above.

For fear that this version of the question would be downvoted and closed (for being imprecise and/or lacking details) I have tried to give such a list of properties down below. Anyone who has an answer to the main question (or even anyone at all) is free to read or not read the further content below. There are several questions formulated in the body of the text below but the main question and most important for me is still the one above. I emphasize that anyone who wishes to answer this question alone is free to ignore the content below in his answer.

Disclaimer: I won't be extremely precise about the set theoretic foundations in the question as I feel these are not the most important points in it (I apologize in advance for any set theoretic mistakes or inaccuracies in the question - my set theory background is not very extensive). In particular I will assume there are enough large cardinals to make everything that follows work.

The question, as in the title, may be very simply stated as follows:

Main Question: Can the homotopy $(2,2)$-category of $(\infty,1)$-categories be characterized as the unique $2$-category upto equivalence satisfying a list of axioms phrased using only $1$-categorical and $2$-categorical concepts?

For fear that this version of the question would be shut down I have tried to give such a list of properties down below. Anyone who has an answer to the main question (or even anyone at all) is free to read or not read the further content below. There are several questions formulated in the body of the text below but the main question and most important for me is still the one above.

Disclaimer: I won't be extremely precise about the set theoretic foundations in the question as I feel these are not the most important points in it (I apologize in advance for any set theoretic mistakes or inaccuracies in the question - my set theory background is not very extensive). In particular I will assume there are enough large cardinals to make everything that follows work.