I am trying to solve 5th problem in Exercises 2.9 in Awodey's book on page 55:
Show that for any boolean algebra $B$, boolean homomorphisms $h : B \to 2$ correspond exactly to ultrafilters in $B$.
I do not understand what the question is asking me to prove because:
a) Boolean homomorphisms in a category $\mathsf{Bool}$ are functors mapping objects of a boolean algebra to objects of some other boolean algebra and then mapping the arrows in a similar fashion.
b) an ultrafilter is a kind of a filter of a boolean algebra - it is a subset of the objects of a boolean algebra.
Therefore how can a function correspond to a set of objects? It cannot be equivalence since the types of the mathematical objects (function, set) are different. What correspondence does Awodey mean here?
By "correspondence" we mean "bijection" in this case. That is, there is a (natural) bijection between the set of ultrafilters of $B$ and the set of Boolean homomorphisms $B\to\textbf{2}$.
Note that a Boolean homomorphism $h:B\to\textbf{2}$ is determined precisely by its kernel in $B$. (Why?) A kernel of a Boolean homomorphism is always an ideal of its domain, but in the case of Boolean homomorphisms into $\textbf{2}$, the kernel is a prime ideal of its domain, and so its complement is an ultrafilter. In this fashion, we find a natural bijection as described.