Given two finite-dimensional vector spaces $V$ and $W$ there is an isomorphism $\mathrm{Hom}(V,W) \cong V^\ast \otimes W$. One way to construct it is to observe that the map $V^\ast \times W \to \mathrm{Hom}(V,W)$ that sends a pair $(\omega,v) \in V^\ast \times W$ to the linear map $ u \mapsto \omega(u)v$ is in fact bilinear, hence, by the universal property of the tensor product, it specifies a unique linear map from $V^\ast \otimes W$ to $\mathrm{Hom}(V,W)$, which can be shown to be an isomorphism.
Now, one could also start with a bilinear map $W \times V^\ast \to \mathrm{Hom}(V,W)$ defined by sending $(v,\omega) \in W \times V^\ast$ to the map $u \mapsto \omega(u)v$. By the same reasoning, this will lift to an isomorphism between $W\otimes V^\ast$ and $\mathrm{Hom}(V,W)$.
Is there anything that would make the same isomorphism somehow more "canonical" than the second, since this is the order in which it always appears in the literature?