 Critical point of an elementary embedding

For an elementary embedding between suitable transitive models of set theory, its critical point of an elementary embedding is the least ordinal $\kappa$ with $j(\kappa )\ne \kappa$, provided such an ordinal exists. Since $j$ preserves ordinal order and fixes every $\alpha <\kappa$, one has $j(\kappa )>\kappa$.

In the large-cardinal setting $j:V_{\lambda }\to M$ with $\lambda$ inaccessible and $M$ transitive, the embedding fixes every element of the full ambient rank $V_{\kappa }$. One way to see the required size control is that $\kappa$ is inaccessible in the domain: each lower-rank set has size less than $\kappa$, so an enumeration of length below $\kappa$ is fixed pointwise once its values are fixed. Induction on rank then fixes all of $V_{\kappa }$.

The reference to the full rank domain matters. An arbitrary inner model may omit ambient sets of rank below $\kappa$, so a claim that its embedding fixes every ambient member of $V_{\kappa }$ would not even be defined on those omitted sets.