The axioms of ETCS can be summed up in one sentence as:
For more details see
Todd Trimble has a series of expositional writings on ETCS which provide a very careful introduction and at the same time a wealth of useful details.
ETCS was proposed by Bill Lawvere, who published an undergraduate set-theory textbook using it:
An informative discussion of the pros and cons of Lawvere’s approach can be found in
Erik Palmgren has a constructive predicative variant of ETCS, which can be summarized as: is a well-pointed -pretopos with a NNO and enough projectives (i.e. COSHEP is satisfied). Here “well-pointed” must be taken in its constructive sense, as including that the terminal object is indecomposable and projective.