stuff
We want to verify the equation
![{\displaystyle \tau _{A}\circ {\mathcal {P}}(f)=f^{*}\circ \tau _{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/573319978e59f2a7ce0be7160f70e6ca48da726c)
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.
![{\displaystyle \tau _{C}(U)=\chi _{U},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa3b6d523502eb1abef004f2dbdff4946d597630)
where χU is given by
![{\displaystyle \chi _{U}(c)={\begin{cases}1&c\in U\\0&c\not \in U\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1c1656d7568bb5e29e1454cd27fa9e08558c3bd)
for any subset U ⊆ C and any element c ∈ C. To verify the equation, let both sides act on some subset S ⊆ B. We have
![{\displaystyle {\mathcal {P}}(f)(S)=f^{-1}(S)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc80160e308bb161264d882ca29de06c68833bb3)
by the definition of the powerset functor, and so
![{\displaystyle \tau _{A}({\mathcal {P}}(f)(S))=\chi _{f^{-1}(S)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0db9e82979126bfeaf99bd0d06eb096b72d3b2c0)
On the right-hand side of the equation, we have
![{\displaystyle \tau _{B}(S)=\chi _{S}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe58f9ef328220c6e08f619afe1346947974b8f9)
and recall that f* is the pullback by f induced by the contravariant hom-functor; it acts on maps by multiplication on the right:
![{\displaystyle f^{*}(\chi _{S})=\chi _{S}\circ f.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d110b60c1a5e00577b3eccb5dde581fb4d4943b)
So it remains to check the equality
![{\displaystyle \chi _{f^{-1}(S)}=\chi _{S}\circ f.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/348e46e1dbf76796b373f5438000285a25b1068b)
To verify this equation, act both maps in 2A on an arbitrary element a ∈ A.
Since a ∈ f–1(S) iff f(a) ∈ S, these maps are equal.
- probably cornbread