On a cofinal Reinhardt embedding without Powerset

In this paper, we provide a positive answer to the question of Matthews whether ${\mathsf{Z}\mathsf{F}}^{-}$ is consistent with a non-trivial cofinal Reinhardt elementary embedding $j\text{\hspace{0.17em}⁣}:V\to V$. The consistency follows from $\mathsf{Z}\mathsf{F}\mathsf{C}+I_0$, and more precisely, it is witnessed by Schlutzenberg’s model of $\mathsf{Z}\mathsf{F}$ with an elementary embedding $k\text{\hspace{0.17em}⁣}:V_{\lambda +2}\to V_{\lambda +2}$.