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.



Weakly initial sets - examples and nonexamples
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. 


