If $\mathcal{C}$ is a skeletally small (i.e. it is equivalent to a small category) preadditive category, then we can make the following construction:


First we form the additive category $\text{Mat} \mathcal{C}$ whose objects are $n$-tuples of objects in $\mathcal{C}$ and whose morphisms between these $n$-tuples are appropriated matrices which entries are morphisms in $\mathcal{C}$. In this case $\mathcal{C}$ can be viewed as a full subcategory of $\text{Mat} \mathcal{C}$ in a canonical way, and $\text{Mat} \mathcal{C}$ is also skeletally small.


Next we use Cauchy Completion to obtain an additive idempotent complete category $\widetilde{\text{Mat} \mathcal{C}}$ in the following way: let $((\text{Mat} \mathcal{C})^{op}, \text{Ab})$ be the (abelian) category of contravariant additive functors from $\text{Mat} \mathcal{C}$ to the category $\text{Ab}$ of abelian groups. Through the Yoneda Embedding we can identify $\text{Mat} \mathcal{C}$ with the full subcategory of $((\text{Mat} \mathcal{C})^{op}, \text{Ab})$ consisting of the representable functors. Then we take $\widetilde{\text{Mat} \mathcal{C}}$ to be full subcategory of $((\text{Mat} \mathcal{C})^{op}, \text{Ab})$ consisting of all retracts of representable functors (in this case these functors will be the direct summands of representable functors). In this case $\widetilde{\text{Mat} \mathcal{C}}$ is the idempotent completion of $\text{Mat} \mathcal{C}$.


Therefore, with this process we obtain an additive idempotent complete category $\widetilde{\text{Mat} \mathcal{C}}$ which canonically contains $\mathcal{C}$ as a full subcategory. Consequently, with this identification we have $\widetilde{\text{Mat} \mathcal{C}} = \text{add} \mathcal{C}$, where $\text{add} \mathcal{C}$ is by definition the full subcategory of $\widetilde{\text{Mat} \mathcal{C}}$ consisting of all direct summands of finite direct sums of objects in $\mathcal{C}$.


Thus this construction give us an additive idempotent complete category $\mathcal{A}$ which contains $\mathcal{C}$ and such that $\text{add} \mathcal{C} = \mathcal{A}$.


**Question:** does this construction appears in the Mathematical literature? If yes, could you give me some references, and maybe explain what is its use?



What is the name of this categorical construction?
This is the [Cauchy completion](https://ncatlab.org/nlab/show/Cauchy+complete+category#InEnrichedCategoryTheory) of $\mathcal{C}$ as an $\mathrm{Ab}$-enriched category.


