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We study functors F from C_f to D where C and D are simplicial modelcategories and C_f is the full subcategory of C consisting of objects thatfactor a fixed morphism f from A to B. We define the analogs of Eilenberg andMac Lane's cross effects functors in this context, and identify explicitadjoint pairs of functors whose associated cotriples are the diagonals of thecross effects. With this, we generalize the cotriple Taylor tower constructionof [10] from the setting of functors from pointed categories to abeliancategories to that of functors from C_f to D to produce a tower of functorswhose n-th term is a degree n functor. We compare this tower to Goodwillie'stower of n-excisive approximations to F found in [8]. When D is stable, and Fis a functor that commutes with realizations, the towers agree. More generally,for functors that do not commute with realizations, we show that the terms ofthe towers agree when evaluated at the initial object of C_f.

Cross effects and calculus in an unbased setting