Example that objects are not all sets in a locally small category

Question

Let $C$ be a locally small category. In this case, is every object set? What would be an example of a locally small category that some objects are not sets?

Answer 1

There are at least very contrived categories like this. Imagine the disjoint union of the objects with a class like $\bf{Ord}$. The morphisms don't need to be set/class functions so give it the same Hom sets.

Answer 2

I think you're not asking the question you mean to. But to literally answer the question...

In whatever foundations you have in mind, is there anything that is not a set?

If so, let $*$ denote such a thing, and let $S$ be any set. Then, we can define a category where

• the only object is $*$
• the only morphism is $S$

Then this category is locally small (because $\{ S \}$ is a set), and none of its objects are sets.

If not, then the objects of any category must be sets, because everything is a set.