Soft robotics is an emerging field of research where the robot body is composed of flexible and soft materials. It allows the body to bend, twist, and deform to move or to adapt its shape to the environment for grasping, all of which are difficult for traditional hard robots with rigid bodies. However, the theoretical basis and design principles for soft robotics are not well-founded despite their recognized importance. For example, the control of soft robots is outsourced to morphological attributes and natural processes; thus, the coupled relations between a robot and its environment are particularly crucial. In this paper, we propose a mathematical foundation for soft robotics based on category theory, a branch of abstract mathematics where any notions can be described by objects and arrows. It allows for a rigorous description of the inherent characteristics of soft robots and their relation to the environment as well as the differences compared to conventional hard robots. We present a notion called the category of mobility to well describe the subject matter. The theory has been applied to a model system and analysis to highlight the adaptation behavior observed in universal grippers, which are a typical example of soft robotics. The aim of the present study is not to offer concrete engineering solutions to existing robotics but to provide clear mathematical description of soft robots by category theory and to imply its potential abilities by a simple soft gripper demonstration. This paper paves the way to developing a theoretical background and design principles for soft robotics.