# Category of objects over $X$

The definition of the category of objects over $X$ is defined as: given a category $C$ and an object $X \in Ob(C)$ the category of objects over $X$ consists of the objects as morphisms $Y \to X$ for some $Y \in Ob(C)$. Morphisms between objects $Y \to X$ and $Y' \to X$ are morphisms $Y \to Y'$ in $C$.

My question is: are $Y$ and $Y'$ different and distinct objects of the category $C$ ? I am sorry that I can't think of a more clear way to state my question, but any help would be much appreciated.

• Yes. ${}{}{}{}$ – Qiaochu Yuan May 14 '14 at 17:32
• @QiaochuYuan Yes?... I really don't see the necessity for $Y$ and $Y'$ of being distinct. – drhab May 14 '14 at 17:40
• I'm not sure what you mean by "distinct." $Y$ and $Y'$ are two different variables which are both separately allowed to vary over all objects. So they are distinct in general but they're allowed to be the same sometimes. – Qiaochu Yuan May 14 '14 at 17:41
• That is a nice clarification of your comment. I think it makes things more clear for the OP. – drhab May 14 '14 at 17:45
• I apologize for the wording of the question but your answers greatly clarify my understanding. - Thanks again – user118822 May 14 '14 at 17:58