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.
- for each morphism $p: s' \to s$ in $P$, the functor $p^* = C(p): C(s) \to C(s')$ has a left adjoint $p_{\sharp}$;
- 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;
- it satisfies the $P$-projection formula: the exchange transformation \begin{aligned} p_\sharp(p^*(-) \otimes -) \implies - \otimes p_\sharp(-) \end{aligned} is an equivalence.
