The MathOverflow question https://mathoverflow.net/questions/338425/a-weak-form-of-shelah-cardinals by Trevor Wilson defines weakly Shelah cardinals as follows:
>A cardinal $\kappa$ is *weakly Shelah* if for all $f : \kappa \to \kappa$ there is some $\alpha < \kappa$ that is closed under $f$ and there is some elementary embedding $j : V \to M$ (where $M$ is a transitive class) such that $\operatorname{crit}(j) = \alpha$ and $j(\alpha) > \kappa$ and $V_{j(f)(\kappa)} \subset M$.

I would like to add a requirement that $j(f) \upharpoonright \kappa = f$ but as a [comment](https://mathoverflow.net/questions/402924/possible-inconsistency-of-weakly-shelah-cardinals-i-hope-not#comment1030681_402924) by Sean Cox on [this question](https://mathoverflow.net/questions/402924/possible-inconsistency-of-weakly-shelah-cardinals-i-hope-not?noredirect=1&lq=1) made me realize, it is not clear that that definition is equivalent to Trevor Wilson's definition. **Are the definitions equivalent?**