If $M$ is a free module of finite rank $n$ over a commutative unitary ring and $a$ is an endomorphism of $M$, consider the endomorphism $\hat a$ of $M$ defined by the identity $$ x_1\wedge ax_2\wedge\dotsb\wedge ax_n =\hat ax_1\wedge x_2\wedge\dotsb\wedge x_n $$ for all $x_1,\dotsc,x_n\in M$. In any basis of $M$, the matrix of $\hat a$ is the adjugate matrix of the matrix of $a$.
Is there a standard name for this "adjugate" endomorphism? Is there a standard notation?
More generally, if $f : M \to N$ is a homomorphism between free modules of rank $n$, then we can define its "adjugate homomorphism" $\overline{f} : N \to M$ to be the composition $$N \xrightarrow{\cong} (\Lambda^{n-1} N)^* \xrightarrow{(\Lambda^{n-1} f)^*} (\Lambda^{n-1} M)^* \xrightarrow{\cong} M.$$ This terminology is used in arXiv:math/9907114, Section 1.2.