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.