Adjoint functors and the classical adjoint

Question

Is there any relationship between adjoint functors seen in category theory, and the classical adjoint (as in adjoint matrices)?

Answer 1

Adjoint functors and adjoint matrices* (and more generally, viewing matrices as linear operators between finite Hilbert spaces, Hermitian adjoints) are both special cases of the notion of adjoint morphism in a bicategory.

Adjoint functors between small categories are precisely adjunctions in the 2-category $\mathbf{Cat}$ of small categories.

On the other hand, we may view any monoidal category $\mathcal V$ as a bicategory $\Sigma \mathcal V$, called the delooping of $\mathcal V$, with a single object $\star$, for which $\Sigma\mathcal V(\star, \star) := \mathcal V$. An adjoint in $\Sigma\mathcal V$ is a dualisable object. Every 2-cell between left adjoints $L \to L'$ in a bicategory induces a 2-cell between their right adjoints $R' \to R$, called its conjugate. Consequently, every morphism between dualisable objects induces a morphism in the opposite direction between their duals, called the dual morphism. Now, for every field $k$, there is a category of vector spaces over $k$, denoted $\mathbf{Vect}_k$. This category is monoidal with respect to the tensor product of vector spaces, and the dualisable objects are precisely the finite vector spaces. For every linear operator $A$ between finite vector spaces, the dual morphism associated to a linear operator $A$ is precisely its adjoint/transpose $A^*$.

This explains the similarity between the two constructions, and justifies the shared terminology.

(*As opposed to adjugate matrices/classical adjoints, which I believe are unrelated to adjoints in category theory.)