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Concrete category is a pair $( \mathcal{C}; U)$ where $\mathcal{C}$ is a category and $U$ is a faithful functor $\mathcal{C} \rightarrow \mathbf{Set}$.

Some concrete categories induce dagger category by the following formula from my draft:

$$f^{\dagger} = \left( U|_{\operatorname{Hom} ( \operatorname{Dst} f ; \operatorname{Src} f)}\right)^{- 1} ( U f)^{- 1}.$$

However this formula may not work in every case. Consider the case of $\operatorname{Hom} ( \operatorname{Dst} f ; \operatorname{Src} f)$ being an empty set.

My question: What additional conditions should we lay on a concrete category for the above formula to characterize a dagger category?

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Well, you certainly need $U(f)$ to be an isomorphism for any $f$ in $\mathcal{C}$. –  roman Oct 22 '13 at 12:06
    
@roman: No, inversion in $(Uf)^{-1}$ means reverse binary relation to the relation $Uf$ not the inverse isomorphism. –  porton Oct 22 '13 at 14:46
    
I see an error in my formula: $(Uf)^{-1}$ not always is a morphisms of $\mathbf{Set}$ (it is however a morphism of $\mathbf{Rel}$). Can my idea be salvaged replacing $\mathbf{Set}$ with $\mathbf{Rel}$? –  porton Oct 22 '13 at 14:47
    
If $U\colon\mathcal{C}\to\mathbf{Rel}$ is also full, it should to work. –  roman Oct 23 '13 at 8:14

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