From Sunder's Functional Analysis: Spectral Theory, Exercise 3.4.3.
Let $\{ \pi_i : A \to \mathcal{L}(\mathcal{H}_i) \}_{i\in I}$ be an arbitrary family of representations of (an arbitrary unital $C^*$-algebra) $A$; show that there exists a unique (unital) representation $\pi : A \to \mathcal{L} (\mathcal{H})$, where $H = \oplus_{i \in I} \mathcal{H}_{i} $, such that $\pi(x) = \oplus_{i \in I} \pi_i(x)$. (See Exercise 2.5.8 for the definition of a direct sum of an arbitrary family of operators.) The representation $\pi$ is called the direct sum of the representations $\{ \pi_i : i \in I\}$, and we write $\pi = \oplus_{i \in I} \pi_i$. (Hint: Use Exercise 2.5.8 and Lemma 3.4.2.)
I do not understand what I am supposed to prove here and how. The exercise given in the hint says there is a unique operator in a direct sum of Hilbert spaces which behaves similarly to the representation given in this problem, I tried to prove this representation is an operator and get the unique operator from that, but I couldn't get anywhere. Also, I'm not certain how to use the lemma, it seems I might have to use it to prove that the norm of the representation matches the one of the operator in the previous exercise, but I also got stuck trying to prove that.