  It is easy to see that for any such embedding Vκ+1⊆ M where κ = crit(j).  This amount of agreement enables one to show that κ is strongly inaccessible, Mahlo, indescribable, ineffable, etc.  To illustrate this let us assume that we have shown that κ is strongly inaccessible and let us show that κ has much stronger large cardinal properties.  Since κ is strongly inaccessible in V and since (Vκ+1)M =Vκ+1, M also thinks that κ is strongly inaccessible.  In particular, M thinks that there is a strongly inaccessible cardinal (namely, κ) below j(κ).  But then by the elementarity of j, V must think the same thing of the preimage of j(κ), namely, κ, that is, V must think that there is a strongly inaccessible below κ.  So κ cannot be the least strongly inaccessible cardinal.  Continuing in this manner one can show that there are many strongly inaccessibles below κ and, in fact, that κ is Mahlo, indescribable, ineffable, etc.  So measurable cardinals subsume the small large cardinals.