Let $\mathcal{C}$ and $\mathcal{D}$ be (small) categories, and let $T:\mathcal{C}\to\mathcal{D}$ and $S:\mathcal{D}\to\mathcal{C}$ be covariant functors. $T$ is said to be a left adjoint functor to $S$ (equivalently, $S$ is a right adjoint functor to $T$) if there is a natural equivalence

Here the functor $\Hom_{{\mathcal{D}}}(T(-),-)$ is a bifunctor $\mathcal{C}\times\mathcal{D}\to\mathbf{Set}$ which is contravariant in the first variable, is covariant in the second variable, and sends an object $(C,D)$ to $\Hom_{{\mathcal{D}}}(T(C),D)$. The functor $\Hom_{{\mathcal{C}}}(-,S(-))$ is defined analogously.

This definition needs additional explanation. Essentially, it says that for every object $C$ in $\cal{C}$ and every object $D$ in $\cal{D}$ there is a function

which is a natural bijection of hom-sets. Naturality means that if $f\colon C^{{\prime}}\to C$ is a morphism in $\mathcal{C}$ and $g\colon D\to D^{{\prime}}$ is a morphism in $\mathcal{D}$, then the diagram

is a commutative diagram. If we pick any $h:T(C)\to D$, then we have the equation

If $T:\mathcal{C}\to\mathcal{D}$ is a left adjoint of $S:\mathcal{D}\to\mathcal{C}$, then we say that the ordered pair $(T,S)$ is an adjoint pair, and the ordered triple $(T,S,\nu)$ an adjunction from $\mathcal{C}$ to $\mathcal{D}$, written

where $\nu$ is the natural equivalence defined above.

An adjoint to a functor is in some ways like an inverse (as in the case of an adjoint matrix); often formal properties about a functor lead to formal properties of its adjoint (for example the right adjoint to a left-exact functor takes injectives to injectives). An adjoint to any functor is unique up to natural isomorphism.

Examples:

1. 1.

   Let $R$ be a commutative ring, and fix an $R$-module $N$. Let

   $${-\otimes N}\colon{R\!-\!\mathbf{mod}}\to{R\!-\!\mathbf{mod}}$$

   be the functor

   $$M\mapsto N\otimes M,$$

   and let

   $${\Hom(N,-)}:{R\!-\!\mathbf{mod}}\to{R\!-\!\mathbf{mod}}$$

   given by

   $$L\mapsto\mathrm{Hom}_{R}(N,L).$$

   Then one can show that ${-\otimes N}$ is the left adjoint to ${\Hom(N,-)}$. This pair of adjoint functors is the most commonly used and studied, and astonishingly deep facts spring from this adjoint relationship.

2. 2.

   Let $U:\mathbf{Top}\to\mathbf{Set}$ be the forgetful functor (i.e. $U$ takes topological spaces to their underlying sets, and continuous maps to set functions). Then $U$ is right adjoint to the functor $F:\mathbf{Set}\to\mathbf{Top}$ which gives each set the discrete topology.

3. 3.

   If $U:\mathbf{Grp}\to\mathbf{Set}$ is again the forgetful functor, this time on the category of groups, the functor $F:\mathbf{Set}\to\mathbf{Grp}$ which takes a set $A$ to the free group generated by $A$ is left adjoint to $U$.

Remarks on Adjointness:

1. 1.

   There are several theorems that link limit and colimit preserving properties of functors to adjointness (e.g., ref. [3]). Thus, a left adjoint functor preserves colimits or acts naturally on the colimit functor (if the latter exists); dually, a right adjoint preserves limits.

2. 2.

   According to William F. Lawvere, Adjointness is closely involved with the Foundation of Mathematics.

3. 3.

   Adjoint functors define dynamic similarities between general systems in categorical dynamics.