I'm trying to learn the basics of homotopy theory by studying from Strom's Modern Classical Homotopy Theory, a book in which the proof of every proposition presented is given as a guided exercise to the reader.
Right now I'm working with homotopy colimits, and I've been stuck with Project 6.26 at page 154 since this morning.
In this problem it is asked to study the set $\mathcal{S}(f)$ of homotopy classes of maps $\Sigma X\rightarrow\Sigma Y$ induced on homotopy colimits by the natural transformation $$ \require{AMScd} \begin{CD} \ast @<<< X @>>> \ast\\ @VVV @V{f}VV @VVV \\ \ast @<<< Y @>>>\ast \end{CD} $$ As suggested by the author, the two maps $\Sigma f$ and $-\Sigma f$ can be obviously found just by taking the cofibrant replacements $$ \require{AMScd} \begin{CD} C_{-}X @<<< X @>>> C_{+}X\\ @V{Cf}VV @V{f}VV @VV{Cf}V \\ C_{-}Y @<<< Y @>>> C_{+}Y \end{CD} $$ and $$ \require{AMScd} \begin{CD} C_{+}X @<<< X @>>> C_{-}X\\ @V{Cf}VV @V{f}VV @VV{Cf}V \\ C_{-}Y @<<< Y @>>> C_{+}Y \end{CD} $$ which may even be in the same homotopy class.
My question is: is there any other element in $\mathcal{S}(f)$? If not, how can I prove that?