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.