For an ideal $mathcal{I}$ of a preadditive category $mathcal{A}$, we study when the canonical functor $mathcal{C}:mathcal{A}{ ightarrow}mathcal{A}/mathcal{I}$ is local. We prove that there exists a largest full subcategory $mathcal{C}$ of $mathcal{A}$, for which the canonical functor $mathcal{C}:mathcal{C}{ ightarrow}mathcal{C}/mathcal{I}$ is local. Under this condition, the functor $mathcal{C}$, turns out to be a weak equivalence between $mathcal{C}$, and $mathcal{C}/mathcal{I}$. If $mathcal{A}$ is additive (with splitting idempotents), then $mathcal{C}$ is additive (with splitting idempotents). The category $mathcal{C}$ is ample in several cases, such as the case when $mathcal{A}$=Mod-R and $mathcal{I}$ is the ideal ${Delta}$ of all morphisms with essential kernel. In this case, the category $mathcal{C}$ contains, for instance, the full subcategory $mathcal{F}$ of Mod-R whose objects are all the continuous modules. The advantage in passing from the category $mathcal{F}$ to the category $mathcal{F}/mathcal{I}$ lies in the fact that, although the two categories $mathcal{F}$ and $mathcal{F}/mathcal{I}$ are weakly equivalent, every endomorphism has a kernel and a cokernel in $mathcal{F}/{Delta}$, which is not true in $mathcal{F}$. In the final section, we extend our theory from the case of one ideal$mathcal{I}$ to the case of $n$ ideals $mathcal{I}_$, ${ldots}$, $mathca{l}_n$.