[Kednicma]

Honestly, it doesn't bother me at all. I'm used to slogans like "syntax and semantics are adjoint functors" or "meaning is a functor from syntax to semantics" from category theory. In the opening definition, for example, I'm reading the LHS as a source category whose objects are abstract syntax trees and whose arrows are substitutions between said trees, and the RHS as any target category of interest. We're giving meanings to notation by sending the notation along a functor, and syntactic/formal manipulations of the notation correspond to manipulations of the objects which they "mean"/represent.

Framed this way, the properties listed are (1) (part of) functorality, (2) surjectiveness, (3/4) smallness, (5) renormalizability (!!), (6) smoothness/continuity of some sort, and (8) the ability to have natural transformations applied. Only (7) is culturally dependent. (1) and (8) are free for all functors!

To address your question on the head (I am not a mathematician), this post gave me some properties of good notation which I hadn't considered before. On one hand, yeah, it's kind of obvious that notation should somehow be able to surjectively reach all possible objects of interest, while still being small enough to definitely not reach everything. On the other hand, renormalization is kind of a dense topic, so it's surprising to see it arise here.