In mathematics, an element x of a star-algebra is self-adjoint if $x^*=x$.
A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if $x^*=y$ then since $y^*=x^{**}=x$ in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements.
In a dagger category, a morphism $f$ is called self-adjoint if $f=f^\dagger$; this is possible only for an endomorphism $f\colon A \to A$.