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 $f$ : $$f(x)=\begin{cases}\ y&\text{if }x\in A\\ \ n&\text{if }x\notin A.\end{cases}$$
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 ?