Weakly initial sets - examples and nonexamples A weakly initial set in a category C is a set of objects I of C such that every object a of C has at least one arrow from an object contained in I.
The question is then, does Fields have a weakly initial set? This is equivalent to the collection of prime fields being a set.
The converse is, is there a (fairly natural) example of a category without a weakly initial set? Aside from obvious things like the discrete category on the objects of a large category.
 A: How about the category of sets with injective maps as morphisms? As an ad hoc example, this may not count as "natural", but it's simple enough. 
[EDIT] following Martin's comment: take the dual, or replace "injective" by "surjective".
A: Regarding the second question: I'm not sure what should count as "natural", but couldn't you just work with examples where the solution set condition in an adjoint functor theorem fails? The solution set is a weakly initial set in a comma category. 
For example, there is no left adjoint to the underlying-set functor $U$ from complete Boolean algebras to sets, and in particular no free complete Boolean algebra on a countably infinite set. But the category of complete Boolean algebras is small-complete and $U$ preserves all small limits. So it's the solution set condition that fails, and therefore the comma category 
$$\mathbb{N} \downarrow U$$ 
has no weakly initial set. 
Edit: After reading David's request for really simple, I offer instead $Ord^{op}$, where $Ord$ is the class of ordinals ordered by inclusion. I acknowledge the influence of Laurent's answer. 
