I am studying Hopf algebras in categories, and I hope, somebody could help me with the following.

Joost Vercruysse in his paper Hopf algebras---Variant notions and reconstruction theorems writes (Theorem 3.6) that the caracterization of Hopf algebras as bialgebras whose category of Hopf modules possesses some "good" properties is generalized to Hopf algebras in "arbitrary" braided categories:

Let $H$ be a bialgebra in a braided monoidal category $C$ with equalizers, then the following statements are equivalent:

(i) the functor $(−)^{coH}:C_H^H\to C$ is fully faithful;

(ii) the pair $(− ⊗ H,(−)^{coH})$ is an equivalence of categories between $C$ and $C_H^H$;

(iii) $(m ⊗ H)\circ (H ⊗ ∆) : H ⊗ H \to H ⊗ H$ is a $C$-isomorphism;

(iv) $H$ admits an antipode, i.e. $H$ is a Hopf algebra in $C$

(here $C_H^H$ is the category of "right-right" Hopf modules over $H$, and $(−)^{coH}$ is the right adjoint functor for $− ⊗ H$).

Vercruysse mentiones this without proof. I think, this is the effect of folklore, when something obvious for specialists is not clear for people who are not aware with the context, but I don't understand how this is proved. Could anybody enlighten me? If this is easy you can just explain this here, but if not, I hope, you could give me a reference.