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@BartoszMilewski you the mathematician must choose which construction you want. We can view any functor $F : C \to Set$ as either covariant in $C$ or contravariant in $C^op$. You can think of this as a "choice of basis/orientation" for the category, and the output of the Grothendieck construction depends on this choice.
The category of pointed sets is not equivalent to the category of sets, but the groupoids are because an iso of pointed sets can't send any of the other points to the base point.
