The assignment $X\mapsto \mathcal{P}\webleft (X\webright )$ together with the functors $f_{!}$, $f^{-1}$, and $f_{*}$ of Item 1 of Proposition 2.6.1.1.4, Item 1 of Proposition 2.6.2.1.3, and Item 1 of Proposition 2.6.3.1.6, and the functors
\begin{align*} -_{1}\cap -_{2} & \colon \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ),\\ \webleft [-_{1},-_{2}\webright ]_{X} & \colon \mathcal{P}\webleft (X\webright )^{\mathsf{op}}\times \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ) \end{align*}
of Item 1 of Proposition 2.3.9.1.2 and Item 1 of Proposition 2.4.7.1.3 satisfy several properties reminiscent of a six functor formalism in the sense of 
.
We collect these properties in Proposition 2.6.4.1.2 below.
