stuff

We want to verify the equation

where τC: P(C) → 2C is the map which sends any subset of the set C to the characteristic function on that subset, i.e.

where χU is given by

for any subset UC and any element cC. To verify the equation, let both sides act on some subset SB. We have

by the definition of the powerset functor, and so

On the right-hand side of the equation, we have

and recall that f* is the pullback by f induced by the contravariant hom-functor; it acts on maps by multiplication on the right:

So it remains to check the equality

To verify this equation, act both maps in 2A on an arbitrary element aA.

Since af–1(S) iff f(a) ∈ S, these maps are equal.

probably cornbread