Lemma 21.2.6. Let \(F\colon C \rightleftarrows D \noloc G\) be an adjunction of \(\infty \)-categories. Then \(F\) preserves all colimits that exist in \(C\) and \(G\) preserves all limits that exist in \(D\).
Proof. We prove that \(F\) preserves colimits; the proof for \(G\) preserving limits is dual. Let \(X\colon I \to C\) be a diagram that admits a colimit in \(C\), and let \(\eta \colon X \to \const _{\colim _I X}\) denote the associated colimit cone. We need to show that the cocone \[ F(\eta )\colon F \circ X \to F(\const _{\colim _IX}) = \const _{F(\colim _I X)} \] is a colimit cone in \(D\). By definition, this means that for every other object \(W\) in \(D\) the map \[ \Hom _D(F(\const _{\colim _IX}),W) \to \Nat (F \circ X,\const _W) \] is an equivalence of animae. But under the adjunction \(F \dashv G\) and the induced adjunction \(F \circ - \dashv G \circ -\) from Lemma 21.1.6, this map is equivalent to the map \[ \Hom _C(\const _{\colim _IX},G(W)) \to \Nat (X, G \circ \const _W) = \Nat (X, \const _{G(W)}). \] Since the latter map is an equivalence by the universal property of the colimit \(\colim _IX\), this finishes the proof. □
Generated from the authoritative LaTeX source.