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For a map f : X → Y , there is the relative model M (Y) = (Λ V , d) → (Λ V ⊗ Λ W , D) ≃ M (X) by Sullivan model theory (Félix et al., Rational homotopy theory, graduate texts in mathematics, Springer, Berlin, 2007). Let Baut 1 X be the Dold–Lashof classifying space of orientable fibrations with fiber X (Dold and Lashof, Ill J Math 3:285–305, 1959]). Its DGL (differential graded Lie algebra)-model is given by the derivations Der M (X) of the Sullivan minimal model M(X) of X. Then we consider the condition that the restriction b f : Der (Λ V ⊗ Λ W , D) → Der (Λ V , d) is a DGL-map and the related topics. [ABSTRACT FROM AUTHOR]