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Objects $X,Y$ with an evaluation morphism $e:X\otimes Y\to I$ and a coevaluation morphism $n:I\to Y\otimes X$ satisfying both snake identities. These are categorical duality data, distinct from the locally convex dual pair.

We use the left internal-hom convention in which $[M,N]$ is right adjoint to $-{\otimes }_{k}M$, so its evaluation morphism has domain $[M,N]{\otimes }_{k}M$. For a Hopf algebra over the commutative ring $k$, take the full $k$-moduleand give it the internal hom for modules over a Hopf algebraThe antipode is an antihomomorphism, $S(hg)=S(g)S(h)$, soThe unit acts identically. Thus this is a left $H$-module.

The evaluation morphism is $H$-linear. Using the diagonal bialgebra action and the antipode identity,Now let $q:L{\otimes }_{k}M\to N$ be $H$-linear and define its ordinary curried map $\hat{q}(\ell )(u)=q(\ell \otimes u)$. To see that it too is $H$-linear, computeConversely, any $H$-linear map $L\to [M,N]$ uncurries to an $H$-linear map by the already established linearity of evaluation. The ordinary tensor–hom adjunction therefore restricts to a natural bijectionPostcomposition by an $H$-linear map makes $[M,-]$ a functor. This proves that the category of left $H$-modules is a left closed monoidal category. All modules are allowed; no inverse antipode or finite-dimensional dual is required. Stating the tensor–hom convention explicitly avoids confusing this construction with the closure on the opposite side.

By the monoidal coherence theorem, calculations may suppress the canonical associators and unitors, restoring them uniquely afterwards. In this notation the two Frobenius monoidal functor identities readWriteWe check both snake identities for this prospective dual pair in a monoidal category. For $FX$, expand and use the first Frobenius identity:The third line uses naturality of $\psi$ and $\phi$. The last line uses $(e\otimes 1)(1\otimes n)=1_X$, followed by the opmonoidal and monoidal unit axioms.

For $FY$, the second Frobenius identity gives the other calculation:Thus both triangular identities hold, andare the evaluation morphism and coevaluation morphism of a dual pair in a monoidal category. Notice that none of the comparison maps was assumed invertible.