Why do dagger categories supposedly capture the structure of a Hilbert space? A dagger functor is a contravariant endofunctor $(\;)^\dagger$ satisfying $X^\dagger = X$ on objects and $f^{\dagger\dagger}$ on morphisms. It is supposed to model adjoint maps on Hilbert spaces, and indeed the category of Hilbert spaces is an excellent example of a dagger category.
It is often claimed, as in this article, that the dagger functor "serves as an axiomatization of the inner product". I have the feeling that there is something fishy about this argument. The standard argument goes like this. Given two vectors $v, w \in \mathcal{H}$, define the functions $f, g: \mathbb{C} \to \mathcal{H}$ by $f(z) = zv$ and $g(z) = zw$, and then set $\left<v|w\right> := f^\dagger g$.
Now, here is what bothers me:

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*This is a definition using the underlying set of $\mathcal{H}$. There is no way of repeating it in an arbitrary dagger category.

*There is no "builtin" guarantee that the defined inner product is actually linear in the second component and antilinear in the first. I need to prove that by hand using linearity of morphism composition and antilinearity of the dagger functor, again a possibly specific feature of Hilbert spaces.

Actually, one can't even say "antilinear" map in an arbitrary dagger category. The right structure is possibly that of an involutive monoidal structure. There, one can axiomatise the concept of the complex conjugate Hilbert space $\overline{\mathcal{H}}$ (the same underlying group, but the complex conjugate scalar multiplication) and finally define an antilinear map $f: \mathcal{H} \to \mathcal{H}$ to be a linear map $f: \mathcal{H} \to \overline{\mathcal{H}}$. An inner product would then be a map $\left<-|-\right>: \overline{\mathcal{H}} \otimes \mathcal{H} \to \mathbb{C}$. But still it's unclear how we could use a dagger structure here to define such a map.
So my question finally is:
What is a categorical way to define the inner product from a dagger structure?
It is clear that some extra structure, for example a monoidal product and probably also the previously mentioned involutive structure is necessary, so in a sense the question is what the necessary extra structure is.
 A: $\mathbb{C}$ is just a particular Hilbert space, and a vector in $H$ is the same thing as a morphism $\mathbb{C} \to H$, so you can talk about these using only the dagger category structure of $\text{Hilb}$, together with the distinguished object $\mathbb{C}$. One reason to distinguish $\mathbb{C}$ is that it's the identity for the tensor product of Hilbert spaces, which is a useful and important extra structure on $\text{Hilb}$, but all you need is $\mathbb{C}$ and this definition reproduces the usual inner product structure on $\text{Hom}(\mathbb{C}, H)$. 

There is no "builtin" guarantee that the defined inner product is actually linear in the second component and antilinear in the first. I need to prove that by hand using linearity of morphism composition and antilinearity of the dagger functor, again a possibly specific feature of Hilbert spaces.

This is a feature, not a bug. This abstract dagger inner product is much more general than the inner product on Hilbert spaces; for example, applied to cobordism categories, it produces an "inner product" on all manifolds with the same fixed boundary $M$ given by gluing together along the common boundary. The output of this process is a manifold (without boundary), but now the cool thing is that you can ask for TQFTs that respect this structure, sending manifolds to Hilbert spaces and cobordisms to linear maps between them in a way that respects dagger structures, and this abstract inner product gets sent to the usual Hilbert space inner product. 
