Warning 8.1.0.4 (Untwisted Arrow Categories). Let $[1] = \{ 0 < 1 \} $ denote a linearly ordered set with two elements. For any category $\operatorname{\mathcal{C}}$, we can identify morphisms of $\operatorname{\mathcal{C}}$ with functors $F: [1] \rightarrow \operatorname{\mathcal{C}}$. The collection of such functors can be organized into a category $\operatorname{Fun}( [1], \operatorname{\mathcal{C}})$, which we refer to as the arrow category of $\operatorname{\mathcal{C}}$. The arrow category $\operatorname{Fun}( [1], \operatorname{\mathcal{C}})$ has the same objects as the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{C}})$. However, the morphisms are different: if $f: X \rightarrow Y$ and $f': X' \rightarrow Y'$ are morphisms of $\operatorname{\mathcal{C}}$, then morphisms from $f$ to $f'$ in $\operatorname{Fun}( [1], \operatorname{\mathcal{C}})$ can be identified with commutative diagrams
where the horizontal maps are oriented in the same direction.