Sure. In mathematics this is why we often use ":=" for definitions, or we indicate in the surrounding text that the next equation is a definition. That would be helpful.
But even then, you cannot define "C" as "C \otimes C", because the right hand side only makes sense if "C" is already defined. And in math you cannot define something twice. As soon as you defined something, it stays the same in the given context.
It's a type signature, not a formula. Cs are not values.
Think of it as a signature in say Rust
fn coproduct(in: C) -> (C, C)
Also you were talking about my knowledge of Hopf algebras, this is not knowledge of Hopf algebra but quibbling about things that are pretty clear from context.
I don't know what prerequisite knowledge is implied for that paper, but for someone who had to dabble in theoretical econometric and ML papers during my graduate studies it is absolutely not clear what is going on. In my experience, a paper written in such fashion where you don't even define what objects you are working with wouldn't even pass as an acceptable undergraduate paper.
For example, if someone works with a probability space (Ω, F, P), they would state so very clearly in their paper even though it is quite obvious from the notation that it is supposed to be a probability space.
Similarly, if someone writes "Given a symmetric monoidal category C with tensor product ⊗...", it is understandable what is going on, or at least it is understandable what I should look up. But in that paper I have no idea how I am supposed to interpret "C", "⊗" and "=" in such a way that the formulas make sense.
> It's a type signature, not a formula. C is not values.
I am not sure what distinction you are trying to draw, but surely any sequence of symbols that adheres to a given formal grammar is a (well-formed) formula; and surely if by value you mean a mathematical object, types and type signatures are values.
> In my experience, a paper written in such fashion where you don't even define what objects you are working with wouldn't even pass as an acceptable undergraduate paper.
Don't read it, it's apparently too stressful for you. You are right, I assume some knowledge but I don't know if this discourse is productive, you are too hung up on things that don't matter.
C is general. Coproduct is a general construct.
Read the Diaconis paper first, you might get what I'm getting at.
But even then, you cannot define "C" as "C \otimes C", because the right hand side only makes sense if "C" is already defined. And in math you cannot define something twice. As soon as you defined something, it stays the same in the given context.