The Stacks project

Example 4.38.7. Let $\mathcal{C}$ be a category. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Consider the representable presheaf $h_ X = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(-, X)$ (see Example 4.3.4). On the other hand, consider the category $p : \mathcal{C}/X \to \mathcal{C}$ from Example 4.2.13. The fibre category $(\mathcal{C}/X)_ U$ has as objects morphisms $h : U \to X$, and only identities as morphisms. Hence we see that under the correspondence of Lemma 4.38.6 we have

\[ h_ X \longleftrightarrow \mathcal{C}/X. \]

In other words, the category $\mathcal{C}/X$ is canonically equivalent to the category $\mathcal{S}_{h_ X}$ associated to $h_ X$ in Example 4.38.5.

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