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Given a bicategory $C$ and a family $cc{W}$ of arrows of $C$, we give conditions on the pair $(C,cc{W})$ that allow us to construct the bicategorical localization with respect to $cc{W}$ by dealing only with the 2-cells, that is without adding objects or arrows to $C$.We show that in this case, the 2-cells of the localization can be given by the homotopies with respect to $cc{W}$, a notion defined in this article which is closely related to Quillen´s notion of homotopy for model categories but depends only on a single family of arrows. {This localization result has a natural application to the construction of the homotopy bicategory of a model bicategory, which we develop elsewhere, as the pair $(C_{fc},cc{W})$ given by the weak equivalences between fibrant-cofibrant objects satisfies the conditions given in the present article.}