On a cofinal Reinhardt embedding without Powerset

We provide a positive answer to the question of 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 $k \colon V_{\lambda +2}\to V_{\lambda +2}$.