A coreflective subcategory is a full subcategory whose inclusion functor has a right adjoint $R$ (a cofree functor):
The dual concept is that of a reflective subcategory. See there for more details.
(equivalent characterizations)
Given any pair of adjoint functors
the following are equivalent:
The left adjoint $L$ is fully faithful. (In this case $A$ is equivalent to its essential image in $B$ under $L$, a full coreflective subcategory of $B$.)
The unit $\eta : 1_A \to R L$ of the adjunction is a natural isomorphism of functors.
The comonad $(L R, L\eta R,\epsilon)$ associated with the adjunction is idempotent, the left adjoint $L$ is conservative, and the right adjoint $R$ is essentially surjective on objects.
If $S$ is the set of morphisms $s$ in $B$ such that $R(s)$ is an isomorphism in $A$, then $R \colon B \to A$ realizes $B$ as the (nonstrict) colocalization of $B$ with respect to the class $S$.
The right adjoint $R$ is codense.
For proofs, see the corresponding characterisations for reflective subcategories.
Vopěnka's principle is equivalent to the statement:
For $C$ a locally presentable category, every full subcategory $D \hookrightarrow C$ which is closed under colimits is a coreflective subcategory.
This is (AdamekRosicky, theorem 6.28).
the inclusion of Kelley spaces into Top, where the right adjoint “kelleyfies”
the inclusion of torsion abelian groups into Ab, where the right adjoint takes the torsion subgroup.
the inclusion of groups into monoids, where the right adjoint takes a monoid to its group of units.
Lie integration, which constructs a simply connected Lie group from a finite-dimensional real Lie algebra. The coreflector is Lie differentiation (taking a Lie group to its associated Lie algebra), and the counit is the natural map to a given Lie group $G$ from the universal covering space of the connected component at the identity of $G$.
In a recollement situation, we have several reflectors and coreflectors. We have a reflective and coreflective subcategory $i_*: A' \hookrightarrow A$ with reflector $i^*$ and coreflector $i^!$. The functor $j^*$ is both a reflector for the reflective subcategory $j_*: A'' \hookrightarrow A$, and a coreflector for the coreflective subcategory $j_!: A'' \hookrightarrow A$.
Jiri Adamek, Jiří Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series 189
Robert El Bashir, Jiri Velebil, Simultaneously Reflective And Coreflective Subcategories of Presheaves (TAC)