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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