Profinite Structures are Retracts of Ultraproducts of Finite Structures

We establish the following model-theoretic characterization: profinite L-structures, the cofiltered limits of finite L-structures, are retracts of ultraproducts of finite L-structures. As a consequence, any elementary class of L-structures axiomatized by L-sentences of the form ∀~x(ψ0(~x) → ψ1(~x)), where ψ0(~x), ψ1(~x) are existencialpositives L-formulas, is closed under the formation of profinite objects in the category L-mod, the category of structures suitable for the language L and L-homomorphisms.