NL-Complete languages are defined by Log-space reduction, while PSPACE complete languages are defined by poly-time many-to-one reduction.

According to these posts :

Why not polynomial-space reductions for $ PSPACE$ -hardness?

PSpace-completeness under PSpace reductions

Every PSPACE language would be PSPACE-Complete if we defined completeness using a Poly-**SPACE** reduction (instead of a poly-**TIME** reduction).

My question is, why does Log-space reduction doesn’t imply completeness for every $ L \in NL$ , for the same reasoning?

Take any **A,B $ \in$ NL** , and fixed $ y,n$ s.t $ y\in A$ and $ n\notin A$ .

We can define the following log-space reduction **$ f$ ** : $ $ f(x)=\begin{cases}\ y&\text{if }w\in A\ \ n&\text{if }w\notin A.\end{cases}$ $

Just solve A in log-space and let our output be the fixed instance according to the right case.

How come log-space reductions are NOT useless for NL completeness while Pspace reductions are useless for PSPACE completeness? What am I missing ?