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

**Examples:**

- In the category of Finitely-generated-projective $R$-modules, we have that: $M^{\vee}:=Hom_R(M,R)$ satisfies: $Hom_R(M^{\vee},R)\cong M$.
- 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 $E$).
- 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?