Six-Functor Formalisms II : The $\mathrm{\infty }$-categorical compactification

This paper is part of a series of articles in which we reproduce the statements regarding the abstract six-functor formalism developed by Liu-Zheng. In this paper, we prove a theorem, which is an $\mathrm{\infty }$-categorical version for defining the exceptional pushforward functor in an abstract-six functor formalism. The article describes specific combinatorial simplicial sets related to compactifications and pullback squares. This theorem plays a key role in constructing the abstract six-functor formalism, which will be discussed in the forthcoming article.