Grothendieck's axiom states that any set is a member of a Grothendieck universe (i.e. of a set that is closed under the subset, powerset, pairing and family-union relations), or equivalently, that there is a proper class of inaccessible cardinals. Tarski's lemma states that any set is a member of a set that is closed under the subset and the powerset relations and that contains each of its subsets of smaller cardinality than itself.
So the axioms seem to state similar things, but I can't really work out the precise connection between them. Does one imply the other? If not, is it known which one has stronger consistency strength?
Is Tarski's axiom as useful as the axiom of universes for category theory? Does the axiom of universe, like Tarski's axiom, imply AC (over ZF)?