Cylinder, Tensor and Tensor-Closed Module

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