This paper introduces the concept of ``generative midtended cognition'', exploring the integration of generative AI with human cognition. The term "generative" reflects AI's ability to iteratively produce structured outputs, while "midtended" captures the potential hybrid (human-AI) nature of the process. It stands between traditional conceptions of intended creation, understood directed from within, and extended processes that bring exo-biological processes into the creative process. We examine current generative technologies (based on multimodal transformer architectures typical of large language models like ChatGPT), to explain how they can transform human cognitive agency beyond what standard theories of extended cognition can capture. We suggest that the type of cognitive activity typical of the coupling between a human and generative technologies is closer (but not equivalent) to social cognition than to classical extended cognitive paradigms. Yet, it deserves a specific treatment. We provide an explicit definition of generative midtended cognition in which we treat interventions by AI systems as constitutive of the agent's intentional creative processes. Furthermore, we distinguish two dimensions of generative hybrid creativity: 1. Width: captures the sensitivity of the context of the generative process (from the single letter to the whole historical and surrounding data), 2. Depth: captures the granularity of iteration loops involved in the process. Generative midtended cognition stands in the middle depth between conversational forms of cognition in which complete utterances or creative units are exchanged, and micro-cognitive (e.g. neural) subpersonal processes. Finally, the paper discusses the potential risks and benefits of widespread generative AI adoption, including the challenges of authenticity, generative power asymmetry, and creative boost or atrophy.