When dealing with tensor products, is it necessary to check the morphisms are well defined?

Question

For example, if I would like to show that for real vector spaces $U$, $V$ and $W$, $\left(U\oplus V\right)\otimes W \cong\left(U\otimes W \right) \oplus \left(V\otimes W\right)$, is it enough to consider the map $f:\left(U\oplus V\right)\times W \to \left(U\otimes W \right) \oplus \left(V\otimes W\right) $ defined by $f((u,v),w)=(u\otimes w, v\otimes w)$ and extending bilinearity. If we check that it is well defined, then $f$ induces a linear map $\tau :\left(U\oplus V\right)\otimes W \to\left(U\otimes W \right) \oplus \left(V\otimes W\right)$, and then show $\ker(\tau)=\{0\}$?