Lemma 4.24.7. Let $u : \mathcal{C} \to \mathcal{D}$ be a left adjoint to the functor $v : \mathcal{D} \to \mathcal{C}$. Let $\eta _ X : X \to v(u(X))$ be the unit and $\epsilon _ Y : u(v(Y)) \to Y$ be the counit. Then

\[ u(X) \xrightarrow {u(\eta _ X)} u(v(u(X)) \xrightarrow {\epsilon _{u(X)}} u(X) \quad \text{and}\quad v(Y) \xrightarrow {\eta _{v(Y)}} v(u(v(Y))) \xrightarrow {v(\epsilon _ Y)} v(Y) \]

are the identity morphisms.

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