[hgsgm]

Sure, if you create a larger object it can contain two smaller objects.

But that construction is so general that it contains all proofs of the from "There exists X that does not have not property Y" that proceed by constructing something that lacks the property, and stuffs the proof into the diagram.

[abdullahkhalids]

Its not fully general. Many different types of undecidability proofs are basically proofs by diagonalization, but not all are. See some counter examples:

https://mathoverflow.net/q/454105

[srcreigh]

“I genuinely don’t have an operational definition for what it means to use diagonalization” - Terry Tao