TITLE: Is there a way to axiomatize the category of sets and relations?
QUESTION [7 upvotes]: The system of axioms known as ETCS axiomatizes the category of sets and functions. Does anyone know of a way to axiomatize the category (and/or allegory) of sets and relations?

REPLY [5 votes]: From Categories, Allegories by Freyd, Scedrov:

2.414. If $\mathbf{C}$ is a topos, then $\mathscr{Rel}(\mathbf{C})$ is a power allegory. Conversely, if $\mathbf{A}$ is a unitary tabular power allegory, then $\mathscr{Map}(\mathbf{A})$ is a topos.

So a reasonable approach would be to start with "$\mathbf{A}$ unitary tabular power allegory", then translate well-pointed, NNO, and AC into the allegory language.