# Universal six-functor formalism on an $\infty$-category

In the article The Universal Six-Functor Formalism by Brad Drew and Martin Gallauer it is proved that for an ordinary category $$S$$ with a wide subcategory $$P$$ of 'smooth morphisms' containing all isomorphisms and stable under pullbacks along morphisms in $$S$$, there is a universal 'pullback functor' $$S^{\rm op} \to \operatorname{CAlg}(\operatorname{Cat}_{\infty})$$ which can be described as assigning to $$s$$ the category $$P_s$$ of smooth morphisms $$p: s' \to s$$ with target $$s$$, with functoriality in $$s$$ given by taking pullbacks.

Question: Does the same hold if $$S$$ is an $$\infty$$-category?

It seems that the proof uses that $$S$$ is an ordinary category, but I don't understand it enough to tell whether the result breaks down if $$S$$ is an $$\infty$$-category. (My main interest is when $$S$$ is a presheaf $$\infty$$-category.)

Added explanation: a functor $$C: S^{\rm op} \to \operatorname{CAlg}(\operatorname{Cat}_{\infty})$$ is called a pullback formalism by Drew and Gallauer if it is $$P$$-left-adjointable, i.e.

1. for each morphism $$p: s' \to s$$ in $$P$$, the functor $$p^* = C(p): C(s) \to C(s')$$ has a left adjoint $$p_{\sharp}$$;
2. it satisfies $$P$$-base change: for each pullback square $$\require{AMScd}$$ $$\begin{CD} t' @>{f'}>> s' \\ @V{p'}VV @VV{p}V \\ t @>{f}>> s \end{CD}$$ with $$p$$ a morphism in $$P$$, the exchange transformation $$p'_{\sharp}(f')^* \implies f^*p_\sharp$$ is a natural equivalence;
3. it satisfies the $$P$$-projection formula: the exchange transformation \begin{aligned} p_\sharp(p^*(-) \otimes -) \implies - \otimes p_\sharp(-) \end{aligned} is an equivalence.
• The authors posit in their introduction that this should usually be possible. Apr 13, 2021 at 12:26
• Oh, I hadn't seen that! Thanks Apr 13, 2021 at 14:11