Let $ f:X\to Y $ be a map in the pointed category of topological spaces $ Top_* $. And let $ U:Top_*\to Top $ be the "forgetful" functor (which "forgets" the basepoint). We can look at the reduced mapping cylinder $ M_f $ and at the unreduced mapping cylinder $ M_{U(f)} $ in the category $ Top $. Until yesterday, I thought $ M_f $ has the same homotopy type of $ M_{U(f)} $. But, in May's Concise Course, he says "If $X, Y$ are well-pointed, then $ M_f $ has the same homotopy type of $ M_{U(f)} $."

Is this hypothesis necessary?

If it is truly necessary, I want to know where I'm wrong. I thought this: Since $ M_{U(f)} $ is a pushout of a trivial cofibration, we have that $ Y\to M_{U(f)} $ is a trivial cofibration. The same way (or only using a explicit homotopy), we have that $ Y\to M_f $ is a homotopy equivalence. So we have that $ M_{U(f)}\equiv Y\equiv M_f $.

I know that the first statement is right. If there is something wrong, it is in the second statement. I believed that we can factor any function in $ Top _* $ in the same way as in $ Top $, id est, $ f= R\circ j $, where $ j: X\to M_f $ is a cofibration and $ R: M_f\to Y $ is a strong retract. Is it wrong?