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