This paper studies categories and functors in the context of reverse and computable mathematics. In ordinary reverse mathematics, we only focuses on categories whose objects and morphisms can be represented by natural numbers. We first consider morphism sets of categories and prove several associated theorems equivalent to \(\mathrm ACA_{0}\) over the base system \(\mathrm RCA_{0}\) . The Yoneda Lemma is a basic result in category theory and homological algebra. We then develop an effective version of the Yoneda Lemma in \(\mathrm RCA_{0}\) ; as an application, we formalize an effective version of the Yoneda Embedding in \(\mathrm RCA_{0}\) . Products and coproducts are basic notions for defining special categories like semi-additive categories and additive categories. We study properties of products and coproducts of a sequence of objects of categories and provide effective characterizations of semi-additive categories and additive categories in terms of products and coproducts. Finally, we further consider the strength of theorems of category theory that are studied in this paper by methods of higher-order reverse mathematics
