Let us use the notation X [n] to mean the following: If X is a spectrum then X [n] = X ∧n . If X is a space then X [n] = X ∧n /∆ n X where ∆ n X ⊂ X ∧n is the fat diagonal. For r ≤ n, there is a functor K r from the category of Σ n -spectra to the category of Σ r -spectra given byHere X may live either in Sp or Top * depending on the type of functors being considered.It turns out that the constructions K r encode all the information about the comonad ∂ * Φ, at least if we restrict our attention to truncated symmetric sequences. More specifically, for r ≤ s ≤ n there is a natural Σ r -equivariant map δ r,s : K r A n → K r K s A n and for each r, a Σ r -equivariant map ǫ r : K r A r → A r that together reflect the comonad structure on ∂ * Φ. These maps are associative and unital in an appropriate sense. We prove the following result (Lemma 4.8), which encodes the ∂ * Φ-coalgebra structure on ∂ * F in terms of the individual maps δ r,s and ǫ r .Theorem 0.4. Let C be either Top * or Sp. Let F : C −→ Sp be a functor. For each r ≤ n there is a Σ r -equivariant map θ r,n : ∂ r F → K r ∂ n F. Moreover, for each r ≤ s ≤ n, the following diagram commutess θr,n Krθs,n / / δr,sand for each r the following composite is the identityThe Taylor tower of F can then be recovered from the symmetric sequence ∂ * F and the maps θ r,n .Note that (0.3) describes the construction K r only up to homotopy. The choice of model for K r and the maps δ r,s matters, because for Theorem 0.4 one needs a model for which the maps δ r,s are strictly associative and unital. At the same time one would like to have a model that is as such that the following diagrams commute