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?



Does the reduced Mapping cylinder have the same homotopy type of unreduced Mapping cylinder?
I don't trust anything with degenerate basepoints, but I do think you are right about the homotopy type via the parenthetical "(or only using an explicit homotopy)''. However, I was working in compactly generated spaces, denoted $\mathcal{U}$, including the weak Hausdorff property. I don't think you can prove that $Mf$, constructed as usual, is in $\mathcal{U}$ without assuming nondegenerate basepoints. Also, without nondegenerate base points, you cannot be sure that $j$ is a cofibration (at least not in the unbased sense; see p. 56 of Concise). In More Concise, Ponto and I take $\mathcal{T}$ to mean nondegenerately based spaces and $\mathcal{U}\_\*$ to mean based spaces in $\mathcal{U}$. This is justified model theoretically by noting that $\mathcal{T}$ is the full subcategory of $h$-cofibrant objects in $\mathcal{U}\_{\*}$ (that is, cofibrant in the based Hurewicz model structure on $\mathcal{U}\_{\*}$). Incidentally, one reason to start work in $\mathcal{U}\_{\*}$ and not
$Top\_{\ast}$ is that the smash product in $Top\_{\ast}$ is not associative (a published source for a very old counterexample is Parametrized homotopy theory, by Sigurdsson and myself). 


