Cylinder, Tensor and Tensor-Closed Module

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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