There is an adjunction between $L$ and $R$ when:
$$ \text{Hom}(LA,B) \approx \text{Hom}(A,RB) $$
Is there something related we can say when instead we have:
$$ \text{Hom}(B,LA) \approx \text{Hom}(A,RB) $$
Is this just a matter of defining the categories in order to have an adjunction? Does this relation has a special name?
A contravariant adjunction on the right consists of contravariant functors $F : \mathcal{C} \to \mathcal{D}$ and $G : \mathcal{D} \to \mathcal{C}$ and a natural bijection $$\mathcal{D} (Y, F X) \cong \mathcal{C} (X, G Y)$$ where $X$ varies in $\mathcal{C}$ and $Y$ varies in $\mathcal{D}$. Note that we can equally well think of this as an ordinary (or covariant) adjunction $$\mathcal{D}^\mathrm{op} (F X, Y) \cong \mathcal{C} (X, G Y)$$ where now $F$ is regarded as a functor $\mathcal{C} \to \mathcal{D}^\mathrm{op}$ and $G$ is regarded as a functor $\mathcal{D}^\mathrm{op} \to \mathcal{C}$.