The digitization of mental health care holds promises of affordable and ubiquitously available treatment, e.g., with conversational agents (chatbots). While technology can guide people to care for themselves, we examined how people can care for another being as a way to care for themselves. We created a self-compassion chatbot (Vincent) and compared between caregiving and care-receiving conditions. Care-giving Vincent asked participants to partake in self-compassion exercises. Care-receiving Vincent shared its foibles, e.g., embarrassingly arriving late at an IP address, and sought out advice. While self-compassion increased for both conditions, only those with care-receiving Vincent significantly improved. In tandem, we offer qualitative data on how participants interacted with Vincent. Our exploratory research shows that when a person cares for a chatbot, the person's selfcompassion can be enhanced. We further reflect on design implications for strengthening mental health with chatbots.