Before stating my question I would like to provide afew motivating examples:


**Examples:**


1. In the category of Finitely-generated-projective $R$-modules, we have that:
$M^{\vee}:=Hom\_R(M,R)$ satisfies: $Hom\_R(M^{\vee},R)\cong M$.
2. If $G$ is a locally compact abelian topological group over the circle group $T$, then the set:
$G^{\wedge}:=Hom\_{CGrp}(G,T)$ of continuous group homomorphisms into the circle group can be given the structure of a locally compact abelian topological group. Moreover the Pontryagin duality theorem states that:
$Hom\_{CGrp}(G^{\wedge},T)\cong G$ as topological groups (note: here $T$ is the terminal object since all groups are considered over $E$).
3. In the category of Banach spaces over ${\mathbb{R}}$the above construction fails, however fails for $Hom\_{\mathbb{R}}(B,{\mathbb{R}})$ however when restricting our attention to the subcategory of continous linear functionals (CLFs) the $B^{\prime}:=Hom\_{CLFs}(B,{\mathbb{R}})$ satisfies the duality relationship:
$Hom\_{CLFs}(B^{\prime},\mathbb{R})\cong B$ for many Banach spaces $B$.


My goal is to understand when this idea can be extended:
**Question:**
When does there exist a faithful subcategory $\mathfrak{C}$ of a category $\mathfrak{D}$ with terminal object $0$ such that, the Hom functor of $\mathfrak{C}$ is internal and for every object $D$ in $\mathfrak{D}$, when considered in $\mathfrak{C}$ there exists some object $D'\in \mathfrak{D}$ satisfying:
$Hom\_{\mathfrak{C}}(D',0)\cong D$ such that the association $D \mapsto D'$ is functorial?



Creating Duals in A Category
Here is a construction that covers the first example but not, I think, the other two. Suppose $C$ is a [closed](http://ncatlab.org/nlab/show/closed+monoidal+category) symmetric monoidal category with unit object $1$ and that $c$ is a [dualizable object](http://ncatlab.org/nlab/show/dualizable+object) in $C$. Then the dual $c^{\ast}$ can functorially identified with the internal hom $[c, 1]$, where $1$ is the monoidal unit, and in particular we always have the reflexivity condition


$$c \cong [[c, 1], 1]$$


because this always holds for the monoidal dual in a symmetric monoidal category. Examples of such $C$ include the symmetric monoidal categories $\text{Mod}(k)$ of modules over a commutative ring $k$, where the dualizable objects are precisely the finitely presented projective $k$-modules. 


