We continue our study of free-algebra functors from a coalgebraic perspective as begun in [8]. Given a set Σ of equations and a set X of variables, let F_Σ(Ⅹ) be the free Σ-algebra over X and V(Σ) the variety of all algebras satisfying E. We consider the question, under which conditions the Set-functor F_Σ weakly preserves pullbacks, kernel pairs, or preimages [9]. We first generalize a joint result with our former student Ch. Henkel, asserting that an arbitrary Set-endofunctor F weakly preserves kernel pairs if and only if it weakly preserves pullbacks of epis. By slightly extending the notion of derivative Σ' of a set of equations Σ as defined by Dent, Kearnes and Szendrei in [3], we show that a functor F_Σ (weakly) preserves preimages if and only if Σ implies its own derivative, i.e. Σ|- Σ', which amounts to saying that weak independence implies independence for each variable occurrence in a term of V(Σ). As a corollary, we obtain that the free-algebra functor will never preserve preimages when V(Σ) is congruence modular. Regarding preservation of kernel pairs, we show that for n-permutable varieties V(Σ), the functor F_Σ weakly preserves kernel pairs if and only if V(Σ) is a Mal'cev variety, i.e. 2-permutable.
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