# DG functors along which contractions can be lifted

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.