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This paper is the second in a series of two papers about generalizingQuillen's Theorem A to strict $\infty$-categories. In the first one, wepresented a proof of this Theorem A of a simplicial nature, direct but somewhatad hoc. In the current paper, we give a conceptual proof of an$\infty$-categorical nature of the same theorem. This proof is based on thetheory of join and slices for strict $\infty$-categories developed by theauthors in a previous paper, and on a comma construction for strict$\infty$-categories generalizing classical comma categories and Gray's comma2-categories. This $\infty$-categorical comma construction is used by the firstauthor in another paper to prove a generalization of Quillen's Theorem B tostrict $\infty$-categories. We believe that the importance of this commaconstruction in the theory of $\infty$-categories goes far beyond the scope ofhomotopy theory.

A Quillen's Theorem A for strict $\infty$-categories II: the $\infty$-categorical proof