Reflective subcategory

A full subcategory $\mathcal{D}\subseteq \mathcal{C}$ is reflective when its inclusion has a left adjoint $L:\mathcal{C}\to \mathcal{D}$, called the reflector. Equivalently, every object $A$ has a universal morphism $A\to LA$ into an object of $\mathcal{D}$.

A reflective subcategory is left-exact when its reflector preserves finite limits. The fixed objects of a left-exact reflector are closed under finite limits.

The reflector of a reflective subcategory is the left adjoint $L$ to its inclusion. Its unit ${\eta }_A:A\to LA$ is universal among morphisms from $A$ to objects of the subcategory.

For a subobject $m:A^{\prime }\hookrightarrow A$ and a left-exact reflector $L$ with unit $\eta$, define $c_A(A^{\prime })$ by the pullback in a categoryThis operation is monotone, inflationary, idempotent, and stable under pullback. If $A$ is fixed by $L$, then $A^{\prime }$ is fixed by $L$ exactly when $A^{\prime }$ is closed.