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A theorem giving sufficient conditions for a functor to have an adjoint functor. The general adjoint functor theorem uses a complete category, locally small categories, preservation of categorical limits and a solution-set condition. The Special adjoint functor theorem obtains that condition from a well-powered source and a small cogenerating family. Precise hypotheses matter in both forms.

For a ring homomorphism $A\to B$, extension of scalars takes an invertible module to an invertible $B$-module. A direct-summand presentation in a finite free module remains a direct-summand presentation after tensoring. At a prime $\mathfrak{q}\subset B$ contracting to $\mathfrak{p}$, localization gives $B_{\mathfrak{q}}{\otimes }_{A_{\mathfrak{p}}}{L}_{\mathfrak{p}}\cong B_{\mathfrak{q}}$. Compatibility with tensor products, identities and composition makes the Picard group of a ring a covariant functor.

A bifunctor is a functor on the product of two categories. It assigns $F(c,d)$ to each pair of objects and $F(u,v)$ to a pair of morphisms, preserving identity morphisms and composition in a category. Equivalently, it is functorial separately in each argument, with the two actions commuting: $F(1,v)F(u,1)=F(u,1)F(1,v)$ whenever the expressions have the corresponding sources and targets. The product category has pairs as objects, pairs of morphisms as arrows, and componentwise composition in a category. A monoidal tensor product is a bifunctor from a category times itself.

For small categories $\mathcal{C},\mathcal{D}$, a functor $F:\mathcal{C}\to \mathcal{D}$ and $Q:\mathcal{C}\to \mathbf{Set}$, choose an infinite cardinal number $\kappa$ bounding the number of morphisms of $\mathcal{D}$ and the total number of elements of $Q$. Given a natural transformation $\eta :Q\to GF$, generate a subfunctor $G^{\prime }\subseteq G$ from all ${\eta }_c(x)$ by applying every outgoing $\mathcal{D}$-morphism. Each $G^{\prime }(d)$ has at most $\kappa$ elements. Label bounded values by subsets of $\kappa$; only a set of such functors and transformations exists. These factorizations give the solution-set condition for precomposition, whose left adjoint is the left Kan extension.

A reduced contravariant homotopy functor on based CW complexes satisfying the wedge-to-product axiom and the Mayer-Vietoris weak-pullback gluing axiom is represented by $[X,Y{]}_{\ast }$ for a CW complex $Y$. Half-exactness alone is not the full hypothesis. A reduced generalized cohomology theory supplies these axioms; suspension compatibility assembles its representing spaces into an Omega-spectrum.

Every functor $F:\mathcal{C}\to \mathbf{Set}$ on a small category is the colimit of the covariant representable functors $\mathcal{C}(A,-)$ indexed by the opposite of its category of elements. The cocone indexed by $(A,x)$ sends $f:A\to B$ to $F(f)(x)$. By the Yoneda lemma, a compatible cocone into $G$ is exactly a natural transformation $F\to G$. The opposite indexing direction is essential because precomposition reverses the representing-object arrow.

Objects are abelian groups and morphisms are group homomorphisms. Addition on hom-sets is pointwise, kernels are subgroup kernels, and cokernels are quotients by images. Thus this is an abelian category. Products are Cartesian products; coproducts are direct sums of finite-support families. The underlying-set functor creates directed colimits, giving a concrete description of their elements by eventual equality of representatives.

Fix a monad on $\mathcal{C}$. Objects are adjunctions with left-hand category $\mathcal{C}$ whose induced monads are identified with the fixed monad. A morphism between two such adjunctions is a functor between their right-hand categories commuting with both adjoints and respecting the specified units and counits. Strict identifications give a category; coherent identifications give the analogous universal property up to compatible natural isomorphism.

Objects are small categories and morphisms are functors. Its products pair objects and arrows componentwise. Its exponential objects are functor categories, as shown by the currying adjunction for small categories.

Category theory studies mathematical structures through the morphisms between them. A category specifies which objects of a category and arrows are allowed; a functor translates these objects and arrows into another category; a natural transformation compares two such translations coherently. For example, the category of groups records both groups and group homomorphisms, so it remembers which maps respect multiplication.

The usefulness appears in constructions we already know. A Cartesian product combines two sets, while a direct product of groups combines two groups. Both satisfy the same universal property of a categorical product: any pair of maps into the factors comes from exactly one map into the product. This gives a reusable way to construct maps and establish isomorphisms, without depending on a particular choice of coordinates or element labels. In a partially ordered set regarded as a category, the corresponding categorical product is a greatest lower bound. The shared definition exposes a connection between pairing data and finding the best common bound.

Start with the computed examples under category, functor, natural transformation, and product in a category. Their arrow equations are practical tests: do both routes in a commutative diagram produce the same result? Later, the same language organizes tensor products, adjunctions, and the fundamental group of a topological space. Emily Riehl's textbook develops these basic notions before using them to compare constructions across different subjects. ^[ref]

In their 1945 paper, Samuel Eilenberg and Saunders Mac Lane set out the notions of category, functor, and natural transformation. The introduction uses the relation between a vector space and its dual space to ask which comparisons are natural, setting up a precise language for compatibility with maps. ^[ref]

A functor between coherent categories is coherent when it preserves finite limits, image factorizations and finite unions of subobjects, including the empty union. It consequently preserves the interpretation of every coherent formula. A coherent functor from a coherent syntactic category to a topos is equivalent to a model of the corresponding coherent theory; natural transformations correspond to ordinary model homomorphisms.

For small $J,K$ and a functor $D:J\times K\to \mathcal{C}$ into a complete category, the two iterated categorical limits are canonically isomorphic. They have the same projections to every $D(j,k)$ and the same universal property. The result follows also from preservation of categorical limits by the limit functor.

Two objects of a category lie in the same connected component if a finite zigzag of morphisms, with either orientation allowed, joins them. This is an equivalence relation; its classes give full subcategories, each a connected category. For a small category, the collection of components is a set denoted ${\pi }_0(\mathcal{C})$. A functor induces a function on components. Empty categories have no components.

For a functor $F:\mathcal{C}\to \mathcal{D}$, precomposition by $u:B\to B^{\prime }$ induces $(B^{\prime }\downarrow F)\to (B\downarrow F)$, hence a map of categorical connected components in the opposite direction to $u$. This defines a presheaf on $\mathcal{D}$. Its category of elements is the intermediate category in the final-discrete-fibration factorization.

A functor is conservative when it reflects isomorphisms: if $F(f)$ is invertible then $f$ is invertible. It need not be full or faithful functor. The crude monadicity theorem uses this reflection property to lift invertibility of the comparison-adjunction counit from the base category.

If some $f_i$ is not injective, extend two distinct equal-image elements to tuples using common choices at all other coordinates; the product is not injective. If some $f_i$ is not surjective, extend a missing target element to a tuple; the product is not surjective. The axiom of choice supplies these extensions. Thus the set-product functor reflects isomorphisms between families whose coordinates are all nonempty.

The constant diagram functor sends $C$ to the functor on $J$ with constant object value $C$ and identity arrow values; a morphism $f$ becomes the natural transformation with every component $f$. When all $J$-shaped categorical limits exist, it is a left adjoint to the limit functor.