tensor products - The meaning of an intertwiner?

I think that a reformulation of my question is necessary:
An intertwiner $iota:; V_{j_{1}}bigotimes V_{j_{2}}rightarrow V_{j_{3}}$ is defined as:

$forall gin SU(2),;forall u_{i},v_{i},w_{i},...in V_{j_{i}}:;iota((rho_{j_{1}}bigotimesrho_{j_{2}})(g);[v_{1}bigotimes v_{2}])=rho_{j_{3}}(g)[iota(v_{1}bigotimes v_{2})]$

where $rho_{j_{i}}$ is the representation map corresponding to the irrep of spin $j_i$ of $SU(2)$, and $V_{j_i}$ are the invariant spaces upon which acts the $rho_{j_{i}}(g)$ for $gin SU(2)$

I know that the Shur's lemma is: if

$iota:; V_{j}rightarrow V_{k}$

is an interwtwiner, then is it either a scalar (if $j=k$) or zero ($jnot= k$)

Now, what I want to know, is if $V_{j_{1}}bigotimes V_{j_{2}} = dotsoplus V_{j_{2}} oplus dots$ take an example $V_{j_{1}}bigotimes V_{j_{2}} = V_{j_{4}} oplus V_{j_{2}} oplus V_{j_{5}}$ I can write:

$forall gin SU(2),;forall u_{i},v_{i},w_{i},...in V_{j_{i}}:;iota((rho_{j_{4}}oplusrho_{j_{3}}oplusrho_{j_{5}})(g);[v_{1}bigotimes v_{2}])=rho_{j_{3}}(g)[iota(v_{1}bigotimes v_{2})]$

in this case how to prove that $iota$ is a scalar? (by Shur's lemma)