I stated this as a joke, but if you think about it, could this not be true in some sense..? This is the joke:
"Does the Yoneda Lemma have the universal property of being the unique result in category theory through which every well-defined statement factors?"
I'm still very much a beginner in CT, and have but a vague understanding of what the lemma says, but given the gravitas it is assigned by category theorists, as well as the nature of the statement itself (namely that it 'extrinsically determines all intrinsic properties of any object (of any category)', could there be some truth to it?