Etymology of transpose of morphisms in an adjunction Let $F: \mathbf{C} \to \mathbf{D}$ be left adjoint to $G : \mathbf{D} \to \mathbf{C}$, witnessed by the family of bijections between hom-sets, natural in objects $X, Y$: $$\operatorname{Hom}_{\mathbf{C}}(X,GY) \overset{\psi_{X,Y}}{\longrightarrow} \operatorname{Hom}_{\mathbf{D}}(FX,Y).$$

I've noticed that among some authors, the image $\psi_{X,Y}(f)$ of a map $X \overset{f}{\to} GY$ is called the transpose of $f$. Does this have anything to do with a categorical realization of transpositions in linear algebra? Is there a canonical motivating example for adjoint functors involving transposes of linear operators, involving (say) adjoint representations of Lie groups?
The etymology of the word "adjunction" is the characterization of the adjoint, or transpose, of a linear map between inner product spaces by the formula $\langle Av,w\rangle=\langle v,A^T w\rangle$. I guess that the use of "transpose" to indicate passing a morphisms across adjunction comes out of this history. But this seems to be only a psychological, not a technical, explanation. In particular, it's difficult to realize transposes in linear algebra as adjunctions in category theory.
Let $V$ and $W$ be finite free $R$-modules. Since the transpose reverses multiplication of matrices, evaluating the transpose $f^T$ of a map $f: V \to W$ at an element $(w_1, \dots, w_n)$ of $W$ amounts to precomposing the functional $[w_1, \dots, w_n]$ by $f$. So $f^T$ is just the image of $f$ under $\operatorname{Hom}_{R\text{-}\mathbf{Mod}}(-,R),$ which we'll call $(-)^{T}$.

With a little thought and a not-so-little amount of verification and diagram-chasing, one can see that $(-)^{T}$ is right adjoint to its opposite functor $\big((-)^{T} \big)^{\operatorname{op}}$ as functors $$\left((-)^{T}\right)^{\operatorname{op}} : R\text{-}\mathbf{Mod} \leftrightarrows R\text{-}\mathbf{Mod}^{\operatorname{op}} : (-)^{T}.$$

In particular, this restricts on finite free $R$-modules to an isomorphism of categories.