Let $A$ be a complex associative and unital algebra. Denote by $A\text{-mod}$ the category of finite-dimensional left modules over $A$. This is an Abelian category and we denote by $\text{Ch}_{\ge 0} (A\text{-mod})$ the category of non-negative chain complexes in $A\text{-mod}$.
I now want to relate the Hochschild cohomology of the dg-category $\text{Ch}_{\ge 0} (A\text{-mod})$ to the Hochschild cohomology of $A$. Does the following make sense: As indicated here
mathoverflow.net/questions/189/definition-of-hochschild-cohomology-of-a-dg-or-a-infinity-category
the Hochschild cohomology can be described as the derived natural endotransformations of the identity of $\text{Ch}_{\ge 0} (A\text{-mod})$ (if my interpretation is correct). So the Hochschild cochains are \begin{align} \mathbb{R}\text{Nat} (\text{id}_{\text{Ch}_{\ge 0} (A\text{-mod})}, \text{id}_{\text{Ch}_{\ge 0} (A\text{-mod})} ). \end{align} Btw: What is a good reference which describes Hochschild cohomology or rather Hochschild cochains in precisely this way?
We can now express this by the derived end \begin{align} \int_{X \in \text{Ch}_{\ge 0} (A\text{-mod})} \mathbb{R}\text{Hom} (X,X). \end{align} Is this correct so far? Does the end really go over $\text{Ch}_{\ge 0} (A\text{-mod})$ or just $A\text{-mod}$? If so, why?