I'm trying to understand the presentation of the dual and adjoint in Folland's Real Analysis. Let $H$ be a Hilbert space and $V:H \longrightarrow H^*$ be the standard isomorphism where $Vy(x) = \langle x,y \rangle$. Given $T: H \longrightarrow H$ a linear transformation, dualize $T$ as $T^\dagger: H^* \longrightarrow H^*$ with $T^\dagger(f) = f \circ T$. Folland then defines the adjoint of $T$ as $T^* = V^{-1}T^\dagger V$. If I just write out this definition in a bit of a clunky way I get $$T^*(x) = V^{-1}(T^\dagger (Vx)) = V^{-1}(V(Tx)) = Tx$$
This seems to show $T^* =T$. This shouldn't be true, so what am I messing up?
$T^\dagger(f) = f \circ T$ gives $T^\dagger (Vx)=V(x)\circ T$ which is not the same as $V(T(x))$. (The left side is $ \langle Ty, x \rangle$ and the rigth side is $ \langle y, Tx \rangle$).