functor between powersets, that takes subsets of the codomain of f back to subsets of its domain. The left adjoint of this functor is the existential quantifier and the right adjoint is the universal quantifier .

That is, is a functor that, for each subset , gives the subset given by
:those in the image of under . Similarly, the universal quantifier is a functor that, for each subset , gives the subset given by
:those whose preimage under is contained in .

The more familiar form of the quantifiers as used

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