Corollary 21.5.7. If a functor \(\alpha \colon I \to J\) admits a left adjoint, then \(\alpha \) is a final functor. Dually, if it admits a right adjoint it is an initial functor.
Proof. We need to show that for every object \(j\) the relative slice category \(I_{j/}\) is weakly contractible. Let \(\beta \colon J \to I\) be a left adjoint to \(\alpha \). Then the relative slice category \(I_{j/}\) is equivalent to the slice category \(I_{\beta (j)/}\) of \(I\). But this admits an initial object given by \((\beta (j), \id _{\beta (j)})\), hence is weakly contractible by Lemma 21.5.6. □
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