In online customer service applications, multiple chatbots that are specialized in various topics are typically developed separately and are then merged with other human agents to a single platform, presenting to the users with a unified interface. Ideally the conversation can be transparently transferred between different sources of customer support so that domain-specific questions can be answered timely and this is what we coined as a Bot-Agent symbiosis. Conversation transition is a major challenge in such online customer service and our work formalises the challenge as two core problems, namely, when to transfer and which bot or agent to transfer to and introduces a deep neural networks based approach that addresses these problems. Inspired by the net promoter score (NPS), our research reveals how the problems can be effectively solved by providing user feedback and developing deep neural networks that predict the conversation category distribution and the NPS of the dialogues. Experiments on realistic data generated from an online service support platform demonstrate that the proposed approach outperforms state-of-the-art methods and shows promising perspective for transparent conversation transition.