TITLE: Is ETCS well-founded?
QUESTION [5 upvotes]: I can't find a statement about the axiom of regularity anywhere in treatments of ETCS. Perhaps this is due to the unfortunate clash of terminology with 'foundations'.

REPLY [11 votes]: As Noah suggests, the axiom of regularity doesn't make sense in the language of ETCS.  What we can say is:

To construct a model of ETCS+R from a model of ZFC, one doesn't need regularity.
From a model of ETCS+R, one can construct a model of ZFC with regularity, and also separately models of ZFC with regularity replaced by various anti-foundation axioms.

The composite ETCS+R -> ZFC -> ETCS+R is the identity, as are the ill-founded versions.  And the composite ZFC -> ETCS+R -> ZFC is also the identity, as are ill-founded versions whose anti-foundation axiom is sufficiently strong to characterize possible set-membership diagrams structurally.
Among other places, more details can be found in this paper of mine.