A presheaf on a category C is a functor F: C^op → Set. The category of presheaves [C^op, Set] is a fundamental construction: it is complete, cocomplete, and cartesian closed, making it a topos. Every object A of C determines a representable presheaf Hom(−, A), and the Yoneda embedding y: C → [C^op, Set] sending A to Hom(−, A) is full and faithful, so C embeds as a full subcategory of its presheaf category. The presheaf category can be thought of as the free cocompletion of C—it freely adds all colimits.
Take a small concrete category such as a poset (P, ≤) and write out several presheaves as contravariant functors to Set. Compute the representable presheaves and verify that the Yoneda embedding is injective on objects and morphisms. Then explore a non-representable presheaf and understand why it cannot arise as Hom(−, A) for any A.
The Yoneda lemma you've already mastered tells you that each object A in a category C determines a contravariant functor Hom(−, A): C^op → Set, sending each object X to the set of morphisms X → A, and sending each morphism f: X → Y to the precomposition map (− ∘ f): Hom(Y, A) → Hom(X, A). The Yoneda embedding y: C → [C^op, Set] shows that this assignment is full and faithful — so C embeds into a larger category whose objects are *all* contravariant functors from C to Set. A presheaf on C is simply any such functor: F: C^op → Set. The presheaves include the representable ones (Hom(−, A) for each A) but also many others that don't correspond to any single object of C.
The geometric motivation makes the contravariance feel natural. Take C to be the category Open(X) of open sets of a topological space X, with morphisms being inclusions U ↪ V whenever U ⊆ V. A presheaf F on this category assigns a set F(U) to each open set U — think of it as "local data over U" (functions, sections, observations). When U ⊆ V, the morphism V → U in C^op (remember, morphisms reverse in the opposite category) corresponds to a restriction map F(V) → F(U): data defined on a larger open set can be restricted to a smaller one. This is exactly the covariant direction for data flow — data restricts to smaller sets, which is why the functor must be contravariant on the original category. Every presheaf you encounter in geometry, topology, or algebra has this restriction-map flavor.
The presheaf category [C^op, Set] has remarkable categorical properties: it has all small limits and colimits (computed pointwise), it is cartesian closed (you can form "function presheaves"), and it is a topos — a category with enough structure to do logic and set theory internally. None of this requires C itself to be well-behaved; the presheaf construction freely adds whatever C lacks. The slogan is that [C^op, Set] is the free cocompletion of C: every functor from C into a cocomplete category extends uniquely (up to unique natural isomorphism) through the Yoneda embedding. This makes presheaves the universal device for "adding formal colimits" to C.
Not all presheaves are representable, and this non-representability is important rather than a deficiency. A representable presheaf Hom(−, A) knows exactly where every morphism in C points; a non-representable presheaf can encode "generalized objects" that C doesn't contain. In algebraic geometry, for instance, moduli problems (classify all curves of genus g, all elliptic curves with a level structure) often have no representing object in the category of schemes, but they do define perfectly good presheaves — and the project of sheafification and algebraic spaces is essentially about deciding which of these presheaves are "geometric enough" to count as spaces. Understanding presheaves is thus the entry point not just to sheaf theory but to the modern approach to geometry where spaces are defined by what you can map *into* them.