Multi-source domain adaptation is an important branch of transfer learning. Category shift, a prominent challenge in this field, stems from the mismatch between category distributions in the source and target domains. To address this problem, a category-aware and reweighting-based multi-source domain adaptation algorithm is proposed. The algorithm enhances positive transfer between similar categories through a category-aware strategy and introduces a reweighting moment matching strategy to reduce distribution differences at various levels. Additionally, adaptive weights are constructed using pseudo-labels to effectively mitigate the impact of category shift. Experimental results on the Digits-Five and Office-Caltech10 datasets show that the proposed algorithm achieves classification accuracies of 94.11% and 97.18%, respectively. These results indicate that the proposed algorithm significantly improves accuracy in scenarios with category shift compared to current typical multi-source domain adaptation algorithms.