$q:X\rightarrow C$ has the same homotopy type of $M_f\rightarrow X \rightarrow C_f$, what does this mean?

Question

The book I'm reading says that a sequence $A \rightarrow X \rightarrow C$ (with maps $f$ and $q$ respectively) is a cofibre sequence if $q:X\rightarrow C$ has the homotopy type of $M_f\rightarrow X\rightarrow C_f$, in which $M_f$ and $C_f$ are the mapping cylinder and cone, and $M_f\rightarrow X$ is a homotopy equivalence. I don't understand what it means to say that those two have the same homotopy type (since he is not talking about spaces apparently, but about the maps)