There is a simpler way: say `T:V->X` is a map of inner-product vector spaces.  You can view `V` as a category, where `Hom(v,w)` is just a single real number, the inner product `<v,w>`, and similarly for `X`.

The adjoint T*:X->V satisfies

`<Tv,x>=<v,T*x>`, i.e. `Hom(Tv,x)=Hom(v,T*x)`

, meaning it is a right adjoint to T (in a very strong sense: we have equality of these hom-sets instead of just natural isomorphism).

The triviality of this example reflects the fact that that T and T* are called "adjoint" simply because they belong on opposite sides of a comma :)  

In general, if H is any function of two variables, we can say that g is right adjoint to f "with respect to H" if H(f(a),b)=H(a,g(b)), and say that "adjoint functors" are "adjoint with respect to Hom" (up to natural isomorphism, of course).