Before asking my question, let me introduce the relevant terminology.
Throughout, let $(A, \Delta)$ be a compact quantum group.
Definition: A representation $v$ on the Hilbert space $H$ is an element $v\in M(B_0(H)\otimes A)$ such that $(\text{id}\otimes \Delta)(v) = v_{(12)}v_{(13)}$. Here the subscripts with the brackets denote the leg numbering notation.
Definition: An intertwiner from the representation $(H_1, v_1)$ to the representation $(H_2, v_2)$ is an element in $B(H_1,H_2)$ such that $(x\otimes 1)v_1 =v_2(x \otimes 1).$
Question: How should the multiplications $(x \otimes 1) v_1$ and $v_2 (x \otimes 1)$ be interpreted?
One way of making sense of these multiplications is as follows:
Let $A\subseteq B(K)$ be a faithful and non-degenerate representation, say the universal GNS representation of $A$. Then we have a canonical inclusion $M(B_0(H_1) \otimes K) \subseteq B(H_1 \otimes K)$ and we can interpret $x \otimes 1$ as an operator $H_1 \otimes K \to H_2 \otimes K$ and $v_1$ as an operator $H_1 \otimes K \to H_1 \otimes K$ and we can simply form the composition $(x \otimes 1)v_1$ in $B(H_1 \otimes K, H_2 \otimes K)$. Similarly for the other side.
However, can one give a definition that is "space-free", i.e. does not refer to a choice of faithful and non-degenerate representation?
EDIT: Maybe the following works:
Viewing $B(H_1, H_2)$ as a corner of $B(H_1\oplus H_2)$, we have canonical inclusions $$B(H_1,H_2) \otimes A \subseteq B(H_1\oplus H_2) \otimes A\subseteq M(B_0(H_1 \oplus H_2) \otimes A)$$ and we also have a canonical inclusion $$M(B_0(H_1)\otimes A)\subseteq M(B_0(H_1 \oplus H_2)\otimes A)$$ so we can perform the multiplication in $M(B_0(H_1\oplus H_2)\otimes A)$.
However, also the above does not seem quite satisfactory.