The statement is: If a problem $ P $ is $ mathbf {NP} $-hard, then there is a reduction of $ text {FALSIFIABLE} $ at $ P $. ($ text {FALSIFIABLE} $ being the set of formulas for which there is an assignment that makes the formula false; this is trivially equal to the complement of $ text {TAUT} $.)

**This statement is correct.**

Why? Simply because $ text {FALSIFIABLE} $ is $ mathbf {NP} $-Achevée. You can prove it exactly the same way $ text {SAT} $ is proven to be $ mathbf {NP} $-complete, by reversing only the truth values "true" and "false" (that is, instead of looking for an assignment that makes the formula true, you are looking for an assignment that makes the false formula).