This question seems trivial, so hopefully it will be resolved quickly.
As pointed out in this question on quotient categories and localization, the two constructions are sometimes related, but in general quite different.
Let $C$ be a triangulated subcategory of a triangulated category $D$, and let $$Q: D \to D/C$$ be the Verdier quotient. As I understand it, to show the existence of such a quotient, Verdier constructed it as a localization (see the proof in these notes, for example). But it is still referred to as the Verdier quotient.
My question is Is the Verdier quotient category $D/C$ actually a quotient category of $D$, say in the sense of the Wikipedia article?
The motivation of this question is that I am looking at a particular Verdier quotient and I want to understand certain properties of the hom-sets, and it would be easier (for me) to think about a quotient, rather than a localization for this.