Units and counits of the form
\[ \begin{aligned} \text{id}_{\mathcal{P}\webleft (X\webright )} & \hookrightarrow f^{-1}\circ f_{*},\\ f_{*}\circ f^{-1} & \hookrightarrow \text{id}_{\mathcal{P}\webleft (Y\webright )},\\ \end{aligned} \qquad \begin{aligned} \text{id}_{\mathcal{P}\webleft (Y\webright )} & \hookrightarrow f_{!}\circ f^{-1},\\ f^{-1}\circ f_{!} & \hookrightarrow \text{id}_{\mathcal{P}\webleft (X\webright )}, \end{aligned} \]
having components of the form
\[ \begin{gathered} U \subset f^{-1}\webleft (f_{*}\webleft (U\webright )\webright ),\\ f_{*}\webleft (f^{-1}\webleft (V\webright )\webright ) \subset V, \end{gathered} \qquad \begin{gathered} V \subset f_{!}\webleft (f^{-1}\webleft (V\webright )\webright ),\\ f^{-1}\webleft (f_{!}\webleft (U\webright )\webright ) \subset U \end{gathered} \]
indexed by $U\in \mathcal{P}\webleft (X\webright )$ and $V\in \mathcal{P}\webleft (Y\webright )$.