Remark 4.3.1.11. Let $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. Remark 4.3.1.7, we see that the slice and coslice categories of Construction 4.3.1.8 are can be realized as oriented fiber products: more precisely, we have canonical isomorphisms
\[ \operatorname{\mathcal{C}}_{/F} \simeq \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) } \{ F\} \quad \quad \operatorname{\mathcal{C}}_{F/} \simeq \{ F\} \operatorname{\vec{\times }}_{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}. \]