Recent text-to-image models produce high-quality images, yet text ambiguity hinders precise control when specific styles or objects are required. There have been a number of recent works dealing with learning and composing multiple objects and patterns. However, current work focuses almost entirely on image content, overlooking imaging factors such as camera lens, sensor types, imaging viewpoints, and scenes' domain characteristics. We introduce this new challenge as Imaging Factor Disentanglement and show limitations of current approaches in the regime. We, therefore, propose the new method Multi-factor disentanglement through Textual Inversion (MULTI). It consists of two stages: in the first stage, we learn general factors, and in the second stage, we extract dataset-specific ones. This setup enables the extension of existing datasets and novel factor combinations, thereby reducing distribution gaps. It further supports modifications of specific factors and image-to-image generation via ControlNets. The evaluation on our new DF-RICO benchmark demonstrates the effectiveness of MULTI and highlights the importance of Factor Disentanglement as a new direction of research.