[Context]: The numerous challenges that can hinder software companies from gathering their own data have motivated over the past 15 years research on the use of cross-company (CC) datasets for software effort prediction. Part of this research focused on Web effort prediction, given the large increase worldwide in the development of Web applications. Some of these studies indicate that it may be possible to achieve better performance using CC models if some strategy to make the CC data more similar to the within-company (WC) data is adopted. [Goal]: This study investigates the use of a recently proposed approach called Dycom to assess to what extent Web effort predictions obtained using CC datasets are effective in relation to the predictions obtained using WC data when explicitly mapping the CC models to the WC context. [Method]: Data on 125 Web projects from eight different companies part of the Tukutuku database were used to build prediction models. We benchmarked these models against baseline models (mean and median effort) and a WC base learner that does not benefit of the mapping. We also compared Dycom against a competitive CC approach from the literature (NN-filtering). We report a company-by-company analysis. [Results]: Dycom usually managed to achieve similar or better performance than a WC model while using only half of the WC training data. These results are also an improvement over previous studies that investigated the use of different strategies to adapt CC models to the WC data for Web effort estimation. [Conclusions]: We conclude that the use of Dycom for Web effort prediction is quite promising and in general supports previous results when applying Dycom to conventional software datasets.