For finite-dimensional $V$, $V^*\otimes W$ is the space of linear maps $V\to W$.Suggest a slogan
Let \(k\) be a field and \(V, W\) finite-dimensional \(k\)-vector spaces. The \(k\)-linear map \[ \begin{aligned} \dualof{V} \tensor_k W &\to \Hom_k(V, W) \\ \tilde v \tensor w &\mapsto \big(x \mapsto \tilde v(x)\, w\big) \end{aligned} \] is an isomorphism.
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PR-O5YUI$\dualof{V}\tensor_k W\cong\Hom_k(V,W)$ for finite-dimensional $V$ and $W$variant-of