in the pointed slice category \(\Fun(C,T)_{/P_0F,*}\). Thus \(P_nG\) is a delooping of the \(n\)-th homogeneous layer of \(F\).
Proof
By Corollary 5.100, applying \(P_n\) to the defining pushout square for \(G\) gives a pullback square The lower horizontal map points \(P_nG\) in the slice over \(P_0F\). Pulling this square back along the canonical point \(P_0F\to P_{n-1}F\) identifies the defining pullback for \(D_nF\) with the relative loop object of \(P_nG\). See also [Anel et al. 2018, Corollary 3.5.3].
References
Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal. Goodwillie's calculus of functors and higher topos theory. J. Topol., 11 (4), 1100–1132. 2018.
