adjoint

Let \(\mathcal{C}, \mathcal{D}\) be categories and \(\mathcal{L}: \mathcal{C} \rightarrow \mathcal{D}\) resp. \(\mathcal{R}: \mathcal{D} \rightarrow \mathcal{C}\) be functors Then \(\mathcal{L}, \mathcal{R}\) are said to be adjoints, if if there exists a natural isomorphism, the adjunction \(\varphi\) for the hom-functors

\begin{align*} \varphi: \mathrm{Hom}_{\mathcal{D}}(L(-), -) \rightarrow \mathrm{Hom}_{\mathcal{C}}(-, \mathcal{R}(-)) \end{align*}

see:

• left adjoint
• right adjoint