Inner Product & Linear Map property (from Linear Algebra Done Right, 7C question 4)

Question

Might be a silly question, but I am unsure about how this solution got the equality

$\langle T^*Tv, v \rangle = \langle Tv, (T^*)^*v \rangle$

as in, how we can move the linear map, $T$, around like they did in their answer. Below is question 4, chapter 7C from Sheldon Axler's "Linear Algebra Done Right".

And a solution from linearalgebras.com:

Answer 1

If $S=T^{*}$ and $x=Tv$ then $\langle Sx, v \rangle =\langle x, S^{*}(v) \rangle$ right?