The purpose of this note is to show that, if $mathcal{V}$ is a closed monoidal category, the following three notions are equivalent. (1) Category with $mathcal{V}$-structure and cylinder. (2) Tensored $mathcal{V}$-category. (3) Tensor-closed $mathcal{V}$-module. As an application we will show that, if $mathcal{V}$ is closed and symmetric, then given a category $mathcal{S}$ there is an one-to-one correspondence between the set of $mathcal{V}$-structures with cylinder and path on $mathcal{S}$ introduced by Quillen and the set of closed $mathcal{V}$-module structures on $mathcal{S}$ introduced by Hovey.
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