Premise: The adjectives *intrinsic* and *extrinsic* come  from the Latin *intrinsecus* ("inner") and *extrinsecus* ("outer"),  from the adverbs *intra* resp. *extra*, and the p.p. *secutus* of the verb *sequor* ("follow"). 

In the context of category theory, if we want to play the game of "non-philological (i.e. a posteriori) etymology", it is tempting to refer *follow* to  arrows. I would therefore say that *intrinsic* is a  categorical property of an object, which is stated by means of its only structure and self-maps, while  *extrinsic* is a  categorical property of an object which also depend from other objects and maps with different domain or co-domain. In this sense, compactness is an intrinsic property of topological spaces, while e.g. the homotopy extension property, or being an ANR, are extrinsic properties; in fact, most universal properties are. (But, I repeat, this is just a suggestion).