Eilenberg and Steenrod is great, but if you're not quite ready for algebraic topology, Mac Lane and Birkhoff's _Algebra_ [1] is an introductory abstract algebra

Eilenberg and Steenrod is great, but if you're not quite ready for algebraic topology, Mac Lane and Birkhoff's _Algebra_ [1] is an introductory abstract algebra text that makes well-motivated use of category theory throughout. From the preface: …we hold that the general and abstract ideas needed should grow naturally from concrete instances. With this in view, it is fortunate that we do not need to begin with the general notion of a category. The most basic category is the category whose objects are all sets and whose morphisms are all functions (from one set to another); hence we can start Chapter I with sets — more accurately, with sets, functions, and the composition of functions — as the fundamental materials. On this background, Chapter II introduces the integers as the most basic example of an algebraic system. All the other categories which we need are quite "concrete" ones — each object A in the category is a set (with some structure), and each morphism from an object A to an object B in the category is a function (one which preserves the structure) on the set A to the set B. Hence we can give in Chapter I an easy, explicit definition of a "concrete" category, leaving the full treatment of the more general notion of a category to Chapter XV. In the same spirit, the idea fundamental to the notion of adjoint functor turns out to be the simple one of a "universal" construction. This idea, introduced in Chapter I for sets and in Chapter II for other concrete categories (such as monoids and lattices), is then developed with successive examples throughout the subsequent chapters. [1] https://archive.org/details/algebra00macl https://archive.org/details/algebra00macl