Let $\mathcal{C}$ be a sketetally small pre-additive category. For an object $C \in \mathcal{C}$, define $A:=End_{\mathcal{C}}(C)^{op}$. Define $Mod(\mathcal{C})$ to be the category of contravariant functors from $\mathcal{C}$ to $Ab$ the category of all abelian groups.
There is an evaluation functor $e_C: Mod(\mathcal{C}) \rightarrow Mod(A), M \mapsto M(C)$, where $M(C)$ is viewed as an $A$-module as follows: $a x=M(a)(x)$ for $a \in A$ and $x \in M(C)$. Then
(1) how to get that the evaluation functor is dense (i.e. for each $A$-module $N$, there is a $\mathcal{C}$-module $M$ such that $e_C(M)=M(C) \cong N$)?
(2) if $\mathcal{C}$ has only one object $C$, then how to get the evaluation functor is an equivalence of categories?
(What I have tried： for finitely generated projective $A$-module $Ae$, we can find a $C$-module $M=Hom(-,C)e$ such that $e_C(M)=Ae$. For usual $A$-module $N$, if $N$ has a projective presentation $$P_1 \overset{f}{\rightarrow} P_0 \rightarrow N \rightarrow 0,$$
there are $\mathcal{C}$-modules $M_0,M_1$ such that $e_{C}(M_0)=P_0$ and $e_{C}(M_1)=P_1$ and take $f'：M_1 \rightarrow M_0$ such that $e_C(f')=f$. Then $e_C(Cokerf')=N$. So for each $A$-module $N$, could it has a projective presentation?)
Best Answer
Let me change your notation, $A = \text{End}_\mathcal{C}(c)$. Then consider the category $\text{B}A$ to be the small preadditive category with a signle object, with the algebra $A$ as the algebra of endomorphisms of that signle object. Then you get that there is a fully faithful functor $i : \text{B}A \to \mathcal{C}$, that sends the single object of $\text{B}A$ to $c \in \mathcal{C}$. You can verify that the evaluation functor at $c$ is just the restriction functor $i^* : \text{Mod}_\mathcal{C} \to \text{Mod}_{\text{B}A}$ (Note that the category $\text{Mod}_{\text{B}A}$ is equivalent to the category of right $A$-modules).
This answers the second question, indeed if $\mathcal{C}$ is a category with a signle object $c$, then $i$ is actually an equivalence of categories and so is $i^*$ the evaluation at $c$.
For the first question, note that there is a functor $i_* : \text{Mod}_{\text{B}A} \to \text{Mod}_{\mathcal{C}}$, $$ N \mapsto [d \mapsto \text{Hom}_{\text{Mod}_{\text{B}A}}(\text{Hom}_{\mathcal{C}}(c,d),N)], $$ then we get that $i^* i_*(N) \simeq \text{Hom}_{\text{Mod}_{\text{B}A}}(\text{Hom}_{\mathcal{C}}(c,c),N)\simeq \text{Hom}_{A}(A,N)\simeq N$.