Cazanave proved that the set of naive \(\mathbb {A}^1\) -homotopy classes of endomorphisms of the projective line admits a monoid structure whose group completion is genuine \(\mathbb {A}^1\) -homotopy classes of endomorphisms of the projective line. In this very short note we show that, over a field which is not quadratically closed, such a statement is never true for punctured affine space \(\mathbb {A}^n\hspace{-0.1em}\smallsetminus \{0\}\) for \(n\ge 2\) .
