Source-free unsupervised domain adaptation (SFUDA) has gained significant attention as an alternative to traditional unsupervised domain adaptation (UDA), which relies on the constant availability of labeled source data. However, SFUDA approaches come with inherent limitations that are frequently overlooked. These challenges include performance degradation when the unlabeled target data fails to meet critical assumptions, such as having a closed-set label distribution identical to that of the source domain, or when sufficient unlabeled target data is unavailable-a common situation in real-world applications. To address these issues, we propose an asymmetric co-training (ACT) method specifically designed for the SFFSDA scenario. SFFSDA presents a more practical alternative to SFUDA, as gathering a few labeled target instances is more feasible than acquiring large volumes of unlabeled target data in many real-world contexts. Our ACT method begins by employing a weak-strong augmentation to enhance data diversity. Then we use a two-step optimization process to train the target model. In the first step, we optimize the label smoothing cross-entropy loss, the entropy of the class-conditional distribution, and the reverse-entropy loss to bolster the model's discriminative ability while mitigating overfitting. The second step focuses on reducing redundancy in the output space by minimizing classifier determinacy disparity. Extensive experiments across four benchmarks demonstrate the superiority of our ACT approach, which outperforms state-of-the-art SFUDA methods and transfer learning techniques. Our findings suggest that adapting a source pre-trained model using only a small amount of labeled target data offers a practical and dependable solution. The code is available at https://github.com/gengxuli/ACT.