The right adjoint functor of a functor, if it exists, is one of two best approximations to a weak inverse of that functor. (The other best approximation is the functor's left adjoint, if it exists.) Note that a weak inverse itself, if it exists, must be a right adjoint, forming an adjoint equivalence.
A right adjoint to a forgetful functor is called a cofree functor; in general, right adjoints may be thought of as being defined cofreely, consisting of anything that works in an inverse, regardless of whether it’s needed.
The concept generalises immediately to enriched categories and in 2-categories.
Given posets (or prosets) $C$ and $D$ and a monotone function $U: C \to D$, a right adjoint of $U$ is a monotone function $G: D \to C$ such that
for all $x$ in $C$ and $y$ in $D$.
Given locally small categories $C$ and $D$ and a functor $U: C \to D$, a right adjoint of $U$ is a functor $G: D \to C$ with a natural isomorphism between the hom-set functors
Given $V$-enriched categories $C$ and $D$ and a $V$-enriched functor $U: C \to D$, a left adjoint of $U$ is a $V$-enriched functor $F: D \to C$ with a $V$-enriched natural isomorphism between the hom-object functors
Given categories $C$ and $D$ and a functor $U: C \to D$, a right adjoint of $U$ is a functor $G: D \to C$ with natural transformations
satisfying certain triangle identities.
Given a 2-category $\mathcal{B}$, objects $C$ and $D$ of $\mathcal{B}$, and a morphism $U: C \to D$ in $\mathcal{B}$, a right adjoint of $U$ is a morphism $G: D \to C$ with $2$-morphisms
satisfying the triangle identities.
Although it may not be immediately obvious, these definitions are all compatible.
Whenever $G$ is a right adjoint of $U$, we have that $U$ is a left adjoint of $G$.
right adjoints preserve monomorphisms.
See Galois connection for right adjoints of monotone functions.
See adjoint functor for right adjoints of functors.
See adjunction for right adjoints in $2$-categories.
See examples of adjoint functors for examples.