Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Construction 4.3.1.7 (Slice Categories over Diagrams). Let $\operatorname{\mathcal{K}}$ and $\operatorname{\mathcal{C}}$ be categories. For each object $C \in \operatorname{\mathcal{C}}$, we let $\underline{C}: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ denote the associated constant functor (carrying each object of $\operatorname{\mathcal{K}}$ to the object $C$ and each morphism of $\operatorname{\mathcal{K}}$ to the identity morphism $\operatorname{id}_{C}$). The construction $C \mapsto \underline{C}$ determines a functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})$.

For every functor $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$, we let $\operatorname{\mathcal{C}}_{/F}$ denote the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(\operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})} \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})_{/F}$, where $\operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})_{/F}$ is the slice category of Construction 4.3.1.1. Similarly, we let $\operatorname{\mathcal{C}}_{F/}$ denote the fiber product $\operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}}) } \operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})_{F/}$, where $\operatorname{Fun}( \operatorname{\mathcal{K}}, \operatorname{\mathcal{C}})_{F/}$ denotes the coslice category of Variant 4.3.1.4. We will refer to $\operatorname{\mathcal{C}}_{/F}$ as the slice category of $\operatorname{\mathcal{C}}$ over $F$, and to $\operatorname{\mathcal{C}}_{F/}$ as the coslice category of $\operatorname{\mathcal{C}}$ under $F$.