This is somewhat a question out writing styles. In a paper, I want to introduce and "define" an object $X$ by just stating the universal properties which is satisfies. Logically, this is not quite right, as one should also show existence, but I can just refer to a reference for this. My motivation is that I only need to use the universal property, and not the messy details of the actual construction.
It is a bit hard to talk in generalities, so let's just say that in particular $X$ is a normed vector space, and for some subset $X_0$ of $X$, I state as part of the properties that
- $\|x\|\leq 1$ for each $x\in X_0$.
I then remark that actually, once you have all the properties, it follows that really $\|x\|=1$ for each $x\in X_0$.
As far as I can see, to prove this stronger condition, you actually need some explicit construction of $X$, which does satisfy this stronger condition, and then apply the universal property to conclude that any realisation of $X$ must have this property.
This is logically sound, but from a writing style, it seems a little perverse, as it hides from the reader how to prove the statement.
Thinking more, there is something slightly surprising happening here, in that the universal property implies fact (Y) but the only way to prove fact (Y) is to appeal to an explicit construction.
Does anyone have examples of this phenomena in other writing? Should I be worried about this presentation style?