Understanding the definition of left derived functor

Question

I'm studying homological algebra. But I stuck in a place in Rotman's Advanced Algebra II, 3rd ed. What does $T\hat{f}$ mean in the paragraph? The point made me confused is that $T$ is a functor from $_R\textbf{Mod}$ to $_S\textbf{Mod}$, how can $T$ apply on the chain map $\hat{f}$? Is the symbol the author used problematic? And what does the $T\hat{f}$ really mean in the context?

Answer 1

It is an abuse of notation : $T$ induces an obvious functor from chain complexes to chain complexes, and the author is just using the same notation for the induced functor.