Should you wish to convert your comment into an answer, I would mark this question resolved.

"We have the following lemma, the AU-incarnation of the diagonalization lemma. It is reminiscent of Lawvere’s fixed point theorem." - @godelian, thank you very much, this looks promising. I had read about Joyal's work on arithmetic categories here [0], and Maria Maietti's modernization of them, but did not know if they had yet been used to categorify Goedel's 2nd Incompleteness Theorem. - [0] "Categorical folklore": mathoverflow.net/questions/126513/…