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I'm looking at the proof of Higher Algebra Proposition 6.1.6.27, and in the very first sentence of the proof, Lurie states:

The functor $(F\delta)_{\Sigma_n}$ is n-homogeneous by Proposition 6.1.5.4.

Checking back to 6.1.5.4, I see the statement

Let $C$ be a small $\infty$-category which admits finite colimits and let $D$ be an $\infty$-category which admits finite limits and small filtered colimits. Assume that filtered colimits in $D$ are left exact. Then composition with the Yoneda embedding $j : C \to Ind(C)$ induces a fully faithful functor $$\theta : Exc^n_c (Ind(C),D)) \to Fun(C,D)$$ whose essential image is the full subcategory $Exc^n(C,D) \subset Fun(C,D)$ spanned by the n-excisive functors.

which seems totally irrelevant.

Does anyone know what the correct reference is, or if the reference is correct, how to apply the proposition to achieve the result?

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The correct reference is 6.1.4.14. (And the hypothesis of 6.1.6.27 should refer to countable limits and colimits, rather than finite limits and colimits.)

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