Is there any sense in which the full subcategory of n-excisive functors (on spectra or on pointed spaces) on those functors $F$ which satisfy a set of distinct (modulo homotopy equivalence) equations $\{ F(X_{i})\simeq Y_{i} \}_{i=1}^{n}$ is equivalent to the category on which the functors are defined? The "moduli space" of degree-n polynomials in one real variable with n distinct points determined is equivalent to $\mathbb{R}$. If we underdetermine polynomial functors, do we see a similar phenomenon? It is at least known that n-excisive functors form a topos, so it doesn't seem too far-fetched to hope for this to be the case as well.
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