right 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 \(\mathcal{R}\) is said to be a right adjoint, if there exists an adjunction

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