Suppose I have a category $C$ of sets, and $a,b \in C$. How can I express, in the language of category theory, that $a \in b$? (To clarify: the objects of $C$ are actually sets, and I want to express that $a$ actually is a member of $b$, in the underlying set theory.) I am aware of the construction of an "element of a set" as a morphism $f : 1 \rightarrow b$. But I don't know how to say of an object $a$ that it is an element of $b$.
One idea would be to form the coslice category $1 \downarrow C$ (the category of morphisms $g : 1 \rightarrow x$ in $C$) and assume we have an isomorphism $f : (1 \downarrow C) \rightarrow C$ which does the right thing. As a newbie, I'm not positive that works. But even if it does, it has the disadvantage that we need to introduce the functor $f$. Is there a better way?