On a cofinal Reinhardt embedding without Powerset

Abstract: In this paper, we provide a positive answer to a question by Matthews whether $\mathsf{ZF}^-$ is consistent with a non-trivial cofinal Reinhardt elementary embedding $j\colon V\to V$. The consistency follows from $\mathsf{ZFC} + I_0$, and more precisely, it is witnessed by Schlutzenberg's model of $\mathsf{ZF}$ with an elementary embedding $k\colon V_{\lambda+2}\to V_{\lambda+2}$.