Definition 4.24.1. Let $\mathcal{C}$, $\mathcal{D}$ be categories. Let $u : \mathcal{C} \to \mathcal{D}$ and $v : \mathcal{D} \to \mathcal{C}$ be functors. We say that $u$ is a left adjoint of $v$, or that $v$ is a right adjoint to $u$ if there are bijections

$\mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(u(X), Y) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, v(Y))$

functorial in $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and $Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$.

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