I am confused by the following statement in Awodey's Category Theory p. 35:
Ring homomorphisms $A\to \mathbb Z$ into the initial ring $\mathbb Z$ play an analogous and equally important role [to that of boolean algebra morphisms $B\to \mathbb 2$] in algebraic geometry. They correspond to so-called prime ideals, which are the ring-theoretic generalizations of ultrafilters.
This is particularly confusing to me since there are many rings which seem to have no morphisms to $\mathbb Z$. For example, any field has no unital ring morphisms into $\mathbb Z$ since units are preserved by ring morphisms.