Lemma 21.2.1. Let \(I\) and \(C\) be \(\infty \)-categories and let \(F\colon I \to C\) be a functor.
- (1)
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An object \(W\) in \(C\) equipped with a cone \(\epsilon _F\colon \const _W \to F\) is a limit of \(F\) in \(C\), in the sense of Definition 1.7.1, if and only if \(\epsilon _F\) exhibits \(W\) as a right adjoint object to \(F\) under the functor \(\const \colon C \to \Fun (I,C)\), in the sense of Definition 21.1.3.
- (2)
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Dually, an object \(W\) equipped with a cocone \(\eta _F\colon F \to \const _W\) is a colimit of \(F\) in \(C\) if and only if it exhibits \(W\) as a left adjoint object to \(F\) under \(\const \colon C \to \Fun (I,C)\).
Proof. This is simply a matter of unwinding definitions: the condition in (1) that \(W\) is a right adjoint object to \(F\) under \(\const \) means that for every other object \(X\) of \(C\) the composite \[ \Hom _C(X,W) \xrightarrow {\const } \Nat (\const _X,\const _W) \xrightarrow {\epsilon } \Nat (\const _X,F) \] is an equivalence, where we recall that we write \(\Nat (-,-)\) for the hom anima in \(\Fun (I,C)\). But this is precisely the universal property of the limit from Definition 1.7.1! □
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