Adjoint of operator composition in Hilbert spaces

Question

Let $H,K,L$ are Hilbert spaces, $R \in B(H,K)$ and $T \in B(K,L)$.

Show that $(TR)^\star = R^\star T^\star$ (where $\star$ denotes the adjoint operator).

My attempt:

Let $x\in H$, $y\in K$, $z\in L$

$\langle R(x),y\rangle = \langle x,R^\star(y)\rangle$

and

$\langle T(y),z\rangle = \langle y,T^\star(z)\rangle$

$y=R(x)$, $\langle T(R(x)),z\rangle = \langle x,(TR)^\star(z)\rangle$

But I cannot continue. I think there are some missings or something unnecessary.

Answer 1

$$\langle TR(x)\mid y\,\rangle\: =\: \langle R(x)\mid T^*(y)\rangle \: =\: \langle\,x\mid R^*T^*(y)\rangle\,,$$ and by uniqueness of the adjoint you deduce that $(TR)^*=R^*T^*$.