# Kerodon

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Example 4.3.1.10. Let $[0]$ denote the category having a single object and a single morphism. For any category $\operatorname{\mathcal{C}}$, the diagonal map

$\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( [0], \operatorname{\mathcal{C}}) \quad \quad S \mapsto \underline{S}$

is an isomorphism of categories. It follows that, for any object $S \in \operatorname{\mathcal{C}}$, we have canonical isomorphisms

$\operatorname{\mathcal{C}}_{/S} \simeq \operatorname{\mathcal{C}}_{/ \underline{S}} \quad \quad \operatorname{\mathcal{C}}_{S/} \simeq \operatorname{\mathcal{C}}_{ \underline{S} / }.$

Consequently, we can view Construction 4.3.1.1 and Variant 4.3.1.4 as special cases of Construction 4.3.1.8.