Lemma. no point-surjective morphism [math-0009]

If there exists $B \xrightarrow{f} B$ such that $f \circ b \ne b$ for all $1 \xrightarrow{b} B$, then there has no point-surjective morphism can exist for $A \xrightarrow{g} B^A$.

This is a reversed version of Lawvere's fixed point, no need an extra proof.