No, adjoint functors are not related to the adjoint action of a group on its Lie algebra. They're related to another sort of "adjoint". Every linear map between vector spaces \\(T : V \to W \\) gives rise to a linear map going the other way between the dual vector spaces, \\( T^{\ast} : W^{\ast} \to V^{\ast} \\), and we call \\( T^{\ast} \\) the **adjoint** of \\(T\\).

Anyway, learn about left and right adjoints here and your world will expand in a very interesting way. All the stuff we're talking about here is much more generally applicable than differential geometry.

Anyway, learn about left and right adjoints here and your world will expand in a very interesting way. All the stuff we're talking about here is much more generally applicable than differential geometry.