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Complementing Jason Chen's answer: Assume ZFC+$I_1$ and let $j:V_{\lambda+1}\to V_{\lambda+1}$ be elementary, so $\lambda$ is the sup of the critical sequence of $j$. Then $V_{\lambda}$ models $\mathfrak{ZFC}(\mathsf{SOL})$, but $\mathrm{cof}(\lambda)=\omega$. For suppose $f:V_\alpha\to\lambda$ is cofinal and definable over $V_{\lambda+1}$ from the parameter $p\in V_{\lambda}$. Let $n<\omega$ be such that $\alpha,p\in V_{\mathrm{crit}(j_n)}$ (where $j_n=$ the $n$th iterate of $j$). Note that $j_n\circ f\neq f$, because taking $x\in V_\alpha$ with $f(x)>\mathrm{crit}(j_n)$, we get $j_n(f(x))>f(x)$. But $j_n\circ f=f$ because $j_n:V_{\lambda+1}\to V_{\lambda+1}$ is elementary and $j_n(p,\alpha)=(p,\alpha)$.