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$\newcommand{\A}{\operatorname{A}} \newcommand{\B}{\operatorname{B}} \newcommand{\Cat}{\mathcal{Cat}} \newcommand{\Cart}{\mathcal{Cart}} \newcommand{\C}{\mathbf{C}} \newcommand{\F}{\operatorname{F}} \...

I'm sorry in advance if this question does not belong on this site. I am curious as to what is "really" going on when you have a family of functors indexed by elements in a base category, all of which ...