A category $\mathcal C$ consists of:

• A class of objects $\mathrm{Ob}(\mathcal C)$.
• For each pair $A, B$, a set of morphisms $\mathrm{Hom}(A, B)$.
• An associative composition $\circ$ and an identity $1_A$ for each $A$.

Examples: $\mathbf{Set}$ (sets + functions), $\mathbf{Grp}$ (groups + homomorphisms), $\mathbf{Top}$ (topological spaces + continuous maps), $\mathbf{Vect}_k$ (vector spaces + linear maps), $\mathbf{Cat}$ (categories + functors), and many more.

A functor $F: \mathcal C \to \mathcal D$ maps objects to objects and morphisms to morphisms preserving identities and composition. Example: $\pi_1: \mathbf{Top}_* \to \mathbf{Grp}$ assigns to each pointed space its fundamental group; continuous maps go to homomorphisms.

A natural transformation $\eta: F \Rightarrow G$ between functors gives, for each object $A$, a morphism $\eta_A: FA \to GA$ such that all squares commute. Two functors are naturally isomorphic if all $\eta_A$ are isomorphisms.

Universal properties characterize objects up to unique isomorphism — products, coproducts, limits, colimits, adjunctions. Adjoint functors $(F \dashv G)$ provide systematic translations: $\mathrm{Hom}_{\mathcal D}(FA, B) \cong \mathrm{Hom}_{\mathcal C}(A, GB)$ naturally in $A, B$. Free–forgetful pairs, Stone duality, and Galois connections are all adjunctions.