$C_{1}$-cofiniteness and vertex tensor categories

We first generalize the logarithmic tensor category theory of Huang-Lepowsky-Zhang to the more general case that the module category for a vertex operator algebra $V$ (more generally a M\"{o}bius vertex algebra) might not be closed under the contragredient functor. Then by verifying the assumptions to use this generalization, we obtain that (logarithmic) intertwining operators among $C_{1}$-cofinite grading-restricted generalized $V$-modules satisfy the associativity property (operator product expansion) and the category of $C_{1}$-cofinite grading-restricted generalized $V$-modules has a natural vertex tensor category structure. In particular, this category has a natural braided tensor category structure with a twist.