This question is related to a question lately posted to $\cal MO$, as well as to this and that question, and especially the last where this post can be taken as a remedy of it since it was proved inconsistent.

So similarly, we add two partial unary functions $``j,f"$ to the language of $\sf ZF$.

The question is about if we can add the following on top of axioms of $\sf ZF$ [$j,f$ not used in Replacement nor Separation]?

$\exists \alpha: \text{ limit} (\alpha) \land j: V_{\alpha+1} \rightarrowtail V_\alpha \land f: V_\alpha \equiv rng(j) \\ \forall S: j[[S]]; j^{-1}[[S]] \text { both exist }\\\forall S: f[S]; f^{-1} [S] \text { both exist }\\ \forall S \in V_\alpha \times V_\alpha: j(S)=f[S] $

Where: "$\equiv$" signify "bijection", and $g[S]=\{g(x) \mid x \in S\}, g[[S]]= \{g[x] \mid x \in S\}$.

The first three conditions have already been proved consistent (see here and from existence of external automorphisms) , it's the addition of the fourth that is unsolved?

The rationale for this question is that if the above four conditions are met, then we'll have $j^{-1} \circ f: V_\alpha \to V_{\alpha+1}; x \mapsto j^{-1} (f(x)) $ being a bijective function satisfying the conditions of the first posting, that is: $\forall S: (j^{-1} \circ f)[[S]]; (j^{-1} \circ f)^{-1}[[S]] \text { both exist }$