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Example 4.2.14. Given a category $\mathcal{C}$ and an object $X\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we define the category of objects under $X$, denoted $X/\mathcal{C}$ as follows. The objects of $X/\mathcal{C}$ are morphisms $X\to Y$ for some $Y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Morphisms between objects $X\to Y$ and $X\to Y'$ are morphisms $Y\to Y'$ in $\mathcal{C}$ that make the obvious diagram commute. Note that there is a functor $p_ X : X/\mathcal{C}\to \mathcal{C}$ which simply forgets the morphism. Moreover given a morphism $f : X'\to X$ in $\mathcal{C}$ there is an induced functor $F : X/\mathcal{C} \to X'/\mathcal{C}$ obtained by composition with $f$, and $p_{X'}\circ F = p_ X$.

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