Duals of a finite dimensional eveloping coalgebra Let $C^e$ be the enveloping $k$-coalgebra of a $k$-coalgebra $C$ and denote by ${C^e}^{\star}:=\mathrm{Hom}\,_{k}(C^e,k)$.  
Then is ${C^e}^{\star} \cong {C^{\star}}^e$?
 A: (Note, all homs and tensors will be over $k$)
Let $C$ be a coalgebra.  The enveloping coalgebra of $C$ is $C^e:=C\otimes C^{op}$.  Similarly, we can define the enveloping algebra of an algebra by $A^e:=A\otimes A^{op}$.  We have that op commutes with dualization, in the sense that if $C$ is a coalgebra, then $(C^*)^{op}\cong (C^{op})^*$ as algebras.
Given vector spaces $V, W$, we have a natural map $V^*\otimes W^* \to (V\otimes W)^*$ given sending $\phi \otimes \psi$ to the map $v\otimes w \mapsto (\phi(v)\otimes \psi(w))$.  If $V$ and $W$ are finite dimensional, this map is an isomorphism.  One proof is to take a basis for $V,W$, and then examine the corresponding dual bases.  
However, if $V$ and $W$ are both infinite, the map is no longer an isomorphism.  We can factor the natural map as 
$$\hom(V,k)\otimes \hom(W,k)\to \hom(V,\hom(W,k))\cong \hom(V\otimes W,k).$$  Since the last maps is an isomorphism, the natural map is an isomorphism if and only if the first map is.  We assert that it is not.
Let $Z=\hom(W,k).$  An element of $\hom(V,k)\otimes Z$ is a finite sums of the form $\sum \phi_i \otimes z_i$, which under our map is sent to $v\to \sum \phi(i)v z_i$.  The image of such a map is finite dimensional.  Since $V$ and $Z$ are both infinite dimensional, there are maps in $\hom(V,Z)$ with infinite image, and therefore our map cannot be surjective.

It is worth pointing out that the failure of tensor products to commute with dualization means that while the dual of a coalgebra is an algebra, the dual of an infinite dimensional algebra is NOT in general a coalgebra.  
Suppose we have a map $\Delta:C\to C\otimes C$.  Dualizaing and precomposing with the natural map, we have $C^* \otimes C^* \to (C\otimes C)^* \stackrel{\Delta^*}{\to} C^*$.
However if we have a map $\mu:A\otimes A\to A$, we cannot in general factor $\mu^*:A^*\to (A\otimes A)^*$ through $A^*\otimes A^*$.  
One solution to this is to view dualization not as a controvatiant functor from vector spaces to itself, but rather a pair of controvariant functors between vector spaces and topological vector space with the pro-finite topology. 
