Example 4.2.13. Given a category $\mathcal{C}$ and an object $X\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we define the *category of objects over $X$*, denoted $\mathcal{C}/X$ as follows. The objects of $\mathcal{C}/X$ are morphisms $Y\to X$ for some $Y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Morphisms between objects $Y\to X$ and $Y'\to X$ are morphisms $Y\to Y'$ in $\mathcal{C}$ that make the obvious diagram commute. Note that there is a functor $p_ X : \mathcal{C}/X\to \mathcal{C}$ which simply forgets the morphism. Moreover given a morphism $f : X'\to X$ in $\mathcal{C}$ there is an induced functor $F : \mathcal{C}/X' \to \mathcal{C}/X$ obtained by composition with $f$, and $p_ X\circ F = p_{X'}$.

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