definition. unique functor [nakahira2023diagrammatic, eq. 2.3] [tt-003Z]

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A unique functor, is a functor from a category \({\cal C}\) to the terminal category \(\mathbf {1}\), uniquely determined by mapping all arrows in \({\cal C}\) to the identity arrow \(\mathit {1}_{\mathrm {*}}\) of the unique object \(\mathrm {*}\) in \(\mathbf {1}\).

This functor is often denoted by \(! : {\cal C} \to \mathbf {1}\).

Intuitively, the functor \(!\) acts to erase all information about the input.