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Neural networks are a general framework for differentiable optimization which includes many other machine learning approaches as special cases. In this paper we build a category-theoretic formalism around a neural network system called CycleGAN. CycleGAN is a general approach to unpaired image-to-image translation that has been getting attention in the recent years. Inspired by categorical database systems, we show that CycleGAN is a "schema", i.e. a specific category presented by generators and relations, whose specific parameter instantiations are just set-valued functors on this schema. We show that enforcing cycle-consistencies amounts to enforcing composition invariants in this category. We generalize the learning procedure to arbitrary such categories and show a special class of functors, rather than functions, can be learned using gradient descent. Using this framework we design a novel neural network system capable of learning to insert and delete objects from images without paired data. We qualitatively evaluate the system on the CelebA dataset and obtain promising results.