The Idea of Structure as Data and Conditions

In the 1952 document at http://mathdoc.emath.fr/archives-bourbaki/PDF/nbt_029.pdf the only mathematician "pr\'{e}sent" referenced by first name only is Sammy. I was permitted to audit a graduate course on category theory guided by Sammy at Columbia University in the early 1960s. I recall his insistence that mathematical structure is given by data and conditions. Is that idea implicit or explicit in Bourbaki? Has that idea been superceded? How does it relate to the development of algebraic theories as understood by Lawvere, Linton, Barr-Wells, the Elephant, and so on? Ellis D. Cooper [For admin and other information see: http://www.mta.ca/~cat-dist/ ]