Lemma 21.2.8. Consider \(\infty \)-categories \(C\) and \(I\), and assume that \(C\) admits all \(I\)-indexed limits (resp. colimits).
- (1)
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For every \(\infty \)-category \(E\), the functor category \(\Fun (E,C)\) admits \(I\)-indexed limits (resp. colimits).
- (2)
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For every functor \(E' \to E\), the restriction functor \(\Fun (E,C) \to \Fun (E',C)\) preserves \(I\)-indexed limits (resp. colimits).
- (3)
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In particular, for every object \(X\) of \(E\) the functor \[ \ev _X\colon \Fun (E,C) \to C \] preserves \(I\)-indexed limits (resp. colimits).
Proof. We treat the case for limits; the case for colimits is dual. By Corollary 21.2.2 there is an adjunction \[ \const \colon C \rightleftarrows \Fun (I,C) \noloc \lim _I. \] By Lemma 21.1.6 this adjunction induces an adjunction on functor categories of the form \[ \Fun (E,C) \rightleftarrows \Fun (E,\Fun (I,C)) \simeq \Fun (I,\Fun (E,C)). \] By applying Corollary 21.2.2 another time this says that \(\Fun (E,C)\) admits \(I\)-indexed limits.
Observe that the limit-functor for \(\Fun (E,C)\) is given as the composite \[ \Fun (I,\Fun (E,C)) \simeq \Fun (E,\Fun (I,C)) \xrightarrow {(\lim _I)_*} \Fun (E,C). \] This description makes it clear that precomposition with any functor \(E' \to E\) preserves \(I\)-limits. □
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