Lemma 21.2.4. Let \(i\) be an initial object of an \(\infty \)-category \(I\). Then every \(\infty \)-category \(C\) admits \(I\)-indexed limits, and the limit functor is given by evaluation at \(i\): \[ \lim _I = \ev _i\colon \Fun (I,C) \to C. \] Dually, if \(i\) is terminal in \(I\) then \(C\) admits \(I\)-indexed colimits which are given by evaluating at \(i\).
Proof. By Corollary 21.2.2 it suffices to show that \(\ev _i\) defines a right adjoint to \(\const \colon C \to \Fun (I,C)\). This adjunction may be obtained by considering the adjunction \(p_I\colon I \to * \noloc i\) and passing to functor categories using Lemma 21.1.6.
(An alternative proof would be to cite Lemma 21.5.5.) □
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