On the class of the FNP version of the Hamiltonian Cycle problem
The FNP version of an NP-complete language $L$ asks to solve the following problem on every input $x$:

1) If $x \not\in L$, output "NO".

2) If $x \in L$, output some witness "w" such that a polynomial-time entity, given $(x,w)$, can verify that $x \in L$.

Consider the transformation $T$ of the above scheme: If the FNP machine answered "NO", $T$ answers "NO". Otherwise, $T$ answers "YES".

Obviously $T$ is a Karp-reduction of any NP-complete problem to the corresponding FNP-complete instance.