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Unsupervised domain adaptation (UDA) methods have been broadly utilized to improve the models' adaptation ability in general computer vision. However, different from the natural images, there exist huge semantic gaps for the nuclei from different categories in histopathology images. It is still under-explored how could we build generalized UDA models for precise segmentation or classification of nuclei instances across different datasets. In this work, we propose a novel deep neural network, namely Category-Aware feature alignment and Pseudo-Labelling Network (CAPL-Net) for UDA nuclei instance segmentation and classification. Specifically, we first propose a category-level feature alignment module with dynamic learnable trade-off weights. Second, we propose to facilitate the model performance on the target data via self-supervised training with pseudo labels based on nuclei-level prototype features. Comprehensive experiments on cross-domain nuclei instance segmentation and classification tasks demonstrate that our approach outperforms state-of-the-art UDA methods with a remarkable margin.