Category theory can be used to state formulas in First-Order Logic without\nusing set membership. Several notable results in logic such as proof of the\ncontinuum hypothesis can be elegantly rewritten in category theory. We propose\nin this paper a reformulation of the usual set-theoretical semantics of the\ndescription logic $\\mathcal{ALC}$ by using categorical language. In this\nsetting, ALC concepts are represented as objects, concept subsumptions as\narrows, and memberships as logical quantifiers over objects and arrows of\ncategories. Such a category-theoretical semantics provides a more modular\nrepresentation of the semantics of $\\mathcal{ALC}$ and a new way to design\nalgorithms for reasoning.\n