This work is a development of braids, tensor categories and Yang-Baxter operators. According to Li [Li, F. (1998). Weak Hopf algebras and some new solution of quantum Yang-Baxter equation. J. AIgebra 208:72-100; Li, F. (2000). Solutions of Yang-Baxter equation in endomorphism semigroup and quasi-(co)braided almost bialgebras. Comm. Algebra 28 (5): 2253-2270], it can be seen as a continuation of studying (not necessarily invertible) solutions of the (quantum) Yang-Baxter equation. We firstly introduce the right braid monoids and discuss their properties. Then, we define pre-tensor categories, pre-tensor functors and quasi-braided pre-tensor categories, and investiage their characterizations. Three examples are given from respectively a weak Hopf algebra, a crossed S-set of a Clifford monoid and the (strict) right braid category. Two universalities of the (strict) right braid category are gotten in order to characterize a category of general Yang-Baxter operators and a quasi-braided pre-tensor category. In a pretensor category we build a general centre of a pre-tensor category as a generalization of a centre and show that it is a quasi-braided pre-tensor category. At the end, a categorical interpretation of the quantum quasi-double of a weak Hapf algebra is obtained under a certain condition.