In the category of finite dimensional vector spaces, the dimension of a vector space can be obtained by tracing out the identity. This can be framed in terms of adjunctions: the dimension of a vector space can be obtained by composing unit and counit of the endomorphism monad for that vector space.
There is an obvious categorification of the dimension of a finite dimensional vector space to self-dual adjunctions. My question is, is there a name for such an endomorphism of 0-cells, and is this notion used anywhere other than finite dimensional vector spaces?