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For an object $X$ in a DG category, its contraction is $r \in Hom^{-1}(X,X)$ such that $d(r)=1_X$. Let us say that contractions lift along a DG functor $F: \mathcal{C} \to \mathcal{D}$, if for a contraction $r$ of $F(X)$ there always exists a contraction $r'$ of $X$ such that $F(r')=r$.

Suppose that contractions lift along $F: \mathcal{C} \to \mathcal{D}$. Let $F_*$ be induced DG functor $\operatorname{Mod}\mathcal{C} \to \operatorname{Mod}\mathcal{D}$, and let $\operatorname{pre-tr}(F)$ be induced DG functor $\operatorname{pre-tr}(\mathcal{C}) \to \operatorname{pre-tr}(\mathcal{D})$. Can we deduce that contractions lift along $F_*$ or along $\operatorname{pre-tr}(F)$? Maybe, under some assumptions on $F$ or $\mathcal{C}$ or $\mathcal{D}$?

My interest is because I want to check that some DG functors are Dwyer-Kan fibrations.

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