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I know it's not the question, but, anent now-user76479's / I guess-then Arul's comment and your response, one way to make use of an algebra structure on $W$, if it is present, is to define $V^{\otimes W} = V^{\otimes\operatorname{Hom}_\text{alg}(W, k)}$ (i.e., one ‘factor’ in the tensor product for each element of the set of algebra homomorphisms), where $k$ is the underlying field. Of course this can fail for many interesting $W$, but, if $W$ is an étale algebra, then it works as desired.