Initial and terminal objects can be described as adjoints. Let \({\cal C}\) be a category. There exist the unique functor \(! : {\cal C} \to \mathbf {1}\), and a constant object functor \(X : 1 \to {\cal C}\) for each object \(X\).

A left adjoint to \(!\) is exactly an initial object of \({\cal C}\): \[ \mathrm {0} \dashv \ ! : \mathbf {1} \rightleftarrows {\cal C} \]

Similarly, a right adjoint to \(!\) is exactly a terminal object of \({\cal C}\): \[ ! \dashv \mathrm {1} : {\cal C} \rightleftarrows \mathbf {1} \]