The empty category is the category $\text{Ø}_{\mathsf{cat}}$ where
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Objects. We have
\[ \operatorname {\mathrm{Obj}}\webleft (\text{Ø}_{\mathsf{cat}}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Ø}. \] -
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Morphisms. We have
\[ \operatorname {\mathrm{Mor}}\webleft (\text{Ø}_{\mathsf{cat}}\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Ø}. \] -
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Identities and Composition. Having no objects, $\text{Ø}_{\mathsf{cat}}$ has no unit nor composition maps.
