  This article is first in a series of papers where we reprove the statements
in constructing the Enhanced Operation Map and the abstract six-functor
formalism developed by Liu-Zheng. In this paper, we prove a theorem regarding
constructing functors between simplicial sets using the category of simplices.
We shall reprove the statement using the language of marked simplicial sets and
studying injective model structure on functor categories. The theorem is a
crucial tool and will be used repeatedly in reproving the $\infty$-categorical
compactification and constructing the so called Enhanced Operation Map in the
forthcoming articles.
