Remark 4.3.1.7 (015U): Slice Categories as Oriented Fiber Products

Remark 4.3.1.7 (Slice Categories as Oriented Fiber Products). Let $\operatorname{\mathcal{C}}$ be a category and let $\operatorname{Fun}( [1], \operatorname{\mathcal{C}})$ denote the arrow category of $\operatorname{\mathcal{C}}$, so that the elements $0,1 \in [1]$ determine evaluation functors

\[ \operatorname{ev}_0: \operatorname{Fun}([1], \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}}\quad \quad \operatorname{ev}_1: \operatorname{Fun}([1], \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}}. \]

For each object $S \in \operatorname{\mathcal{C}}$, the slice category $\operatorname{\mathcal{C}}_{/S}$ can be identified with the fiber of the evaluation functor $\operatorname{ev}_{1}$ over $S$, and the coslice category $\operatorname{\mathcal{C}}_{S/}$ can be identified with the fiber of the evaluation functor $\operatorname{ev}_0$ over $S$. That is, we have pullback diagrams

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{S/} \ar [r] \ar [d] & \operatorname{Fun}([1], \operatorname{\mathcal{C}}) \ar [d]^{ \operatorname{ev}_0 } & \operatorname{\mathcal{C}}_{/S} \ar [r] \ar [r] \ar [d] & \operatorname{Fun}([1], \operatorname{\mathcal{C}}) \ar [d]^{\operatorname{ev}_1} \\ \{ S\} \ar [r] & \operatorname{\mathcal{C}}& \{ S\} \ar [r] & \operatorname{\mathcal{C}}. } \]

In other words, we can identify $\operatorname{\mathcal{C}}_{/S}$ with the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ S\} $ of Notation 2.1.4.19 (here we identify the object $S$ with the constant functor $[0] \rightarrow \operatorname{\mathcal{C}}$ taking the value $S$), and $\operatorname{\mathcal{C}}_{S/}$ with the oriented fiber product $\{ S\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$.