A mathematical framework based on category theory is proposed to formally describe and explore procedures of modeling engineering products and processes that comprise operation of a model-oriented enterprise. The framework is intended to provide interoperability across a variety of engineering modeling languages and tools, supplying them with a common abstract foundation capable to represent, generate, and verify diverse design and production knowledge. The framework is leveraged via algebraic representation of product configurations as diagrams in categories with models as objects and descriptions of actions involved into products assembly as morphisms. Relevance of the framework is justified by appealing to systems engineering standards such as IEC 81346. Category theoretical methods for solving direct assembly problems that consist in constructing a product model from a given configuration are presented. Specifically, solutions are obtained via the universal construction called a colimit of a diagram. Much attention is then paid to stating and solving inverse assembly problems that consist in recovery and subsequent optimization of the configuration from the product model and assembly actions. Inverse problem solving is in demand for generative design, viz. an emerging fully automatic product development and manufacturing technology. Example solutions to direct and inverse problems are described in categories that represent two major areas of model-based enterprise operation: solid body geometric modeling of mechanical products and discrete-event simulation of production processes.