In detail, the walking arrow is the category $\mathbb {1}$ where:
- Objects. We have $\text{Obj}\webleft (\mathbb {1}\webright )=\webleft\{ 0,1\webright\} $.
- Morphisms. We have
\begin{align*} \textup{Hom}_{\mathbb {1}}\webleft (0,0\webright ) & = \webleft\{ \text{id}_{0}\webright\} ,\\ \textup{Hom}_{\mathbb {1}}\webleft (1,1\webright ) & = \webleft\{ \text{id}_{1}\webright\} ,\\ \textup{Hom}_{\mathbb {1}}\webleft (0,1\webright ) & = \webleft\{ f_{01}\webright\} ,\\ \textup{Hom}_{\mathbb {1}}\webleft (1,0\webright ) & = \text{Ø}. \end{align*}
- Identities and Composition. The identities and composition of $\mathbb {1}$ are completely determined by the unitality and associativity axioms for $\mathbb {1}$.
