Corollary 21.2.2. Let \(I\) and \(C\) be \(\infty \)-categories. Then \(C\) admits all \(I\)-indexed limits if and only if the functor \[ \const \colon C \to \Fun (I,C), \qquad W \mapsto \const _W \] admits a right adjoint \[ \lim _I\colon \Fun (I,C) \to C. \] Dually, \(C\) admits all \(I\)-indexed colimits if and only if \(\const \colon C \to \Fun (I,C)\) admits a left adjoint \[ \colim _I\colon \Fun (I,C) \to C. \]
Proof. In light of Lemma 21.2.1 this is an immediate consequence of the pointwise criterion for adjoints from Corollary 21.1.5. □
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