I just read the following (paraphrased) definition of universal Turing machine (UTM): “a UTM $M$ has the property that for any other Turing machine $N$ there is an invertible map $f$ from the set of possible states of the tape of $N$ into the set of possible states of the tape $M$ such that if we apply $f$ to an input string of $N$, run the image string on $M$ until it halts then apply the inverse map we get exactly the output of $N$ on that string.”
To me (a category theory novice) this sounds almost like a universal property, so I’m wondering if there is a category where universality as a Turing machine is given by some universal property. I’m guessing one issue is that there should also be some uniqueness statement in the definition, which is noticeably absent.