It is known that one can formulate certain large cardinal axioms (e.g. Vopenka's principle--see Mike Shulman's answer to Harry Gindi's mathoverflow question "Reasons to believe Vopenka's Principle/huge cardinals are consistent") in terms of Category theory.
Can Reinhardt cardinals (and consequently Kunen's inconsistency result regarding them) be formulated in terms of Category theory as well? As a 'soft question' I am wondering if such a formulation might show forth (so to speak) any possible "deep inconsistency" (to use Peter Koellner's term) regarding the hypothesis that $ZF$$+$$\exists$$a$ $Reinhardt$ $cardinal$$ZF+\text{ exists a Reinhardt cardinal}$ is consistent?
