NP-Completeness reduction, using a same input
You can do whatever you want as long as you satisfy the definitions.

For example, a polynomial time reduction from a language $A$ to a language $B$ is a function $f$ satisfying the following two properties:


$f$ can be computed in polynomial time.
For every $x$, we have $x \in A$ iff $f(x) \in B$.


Any function $f$ satisfying these conditions is a polynomial time reduction from $A$ to $B$.

In particular, if $A$ is NP-hard and there exists a function $f$ satisfying these two conditions, then $B$ is also NP-hard.

The definition doesn't specify anything beyond these two properties. In mathematics nothing is left unsaid; everything which is not forbidden is actually allowed. All the cards are open. The rules of the game are public, and there is no need to ask a judge about any of them.