Fibred categories of subsets and slices over Set

Each (function $\functionf:\setX\to\setY$) in $\Set$ (the large 1-category of small sets)
generates (the following diagram of categories and functors) in $\CAT$ (the very large 2-category of large categories).
Note: below (the diagram in $\CAT$) we show (the function $\functionf:\setX\to\setY$ in $\Set$) that generates it. $$\begin{array}{cccl} \Set\downarrow\setX & \begin{array}{} \xrightarrow{\textstyle \;\;\sum_\functionf = \functionf_!\;\;}\\ \xleftarrow{\textstyle \functionf^\star}\\ \xrightarrow{\textstyle \prod_\functionf = \functionf_\star}\\ \end{array} & \Set\downarrow\setY \mkern2em & \text{functors between categories of oversets} \\ \llap{\sigma_\setX}\Bigg\downarrow \dashv \Bigg\uparrow \rlap{i_\setX} && \llap{\sigma_\setY} \Bigg\downarrow \dashv \Bigg\uparrow \rlap{i_\setY} & \text{reflections of oversets (supersets) into subsets} \\ \Sub(\setX) & \begin{array}{} \xrightarrow{\textstyle \;\;\exists_\functionf = \functionf_!\;\;}\\ \xleftarrow{\textstyle \functionf^\inv}\\ \xrightarrow{\textstyle \forall_\functionf = \functionf_\star}\\ \end{array} & \Sub(\setY) & \text{order-preserving maps between posets of subsets} \\ \\ \\ \setX & \xrightarrow[\textstyle \functionf]{} & \setY & \text{an arrow (function) in $\Set$} \\ \end{array}$$ Each horizontal triple of arrows, upper and lower, is an adjoint triple, i.e. an adjoint string of length 3.