  Given any additive category $\mathcal{C}$ with split idempotents,
pseudokernels and pseudocokernels, we show that a subcategory $\mathcal{B}$ is
coreflective if, and only if, it is precovering, closed under direct summands
and each morphism in $\mathcal{B}$ has a pseudocokernel in $\mathcal{C}$ that
belongs to $\mathcal{B}$. We apply this result and its dual to, among others,
preabelian and pretriangulated categories. As a consequence, we show that a
subcategory of a preabelian category is coreflective if, and only it, it is
precovering and closed under taking cokernels. On the other hand, if
$\mathcal{C}$ is pretriangulated with split idempotents, then a subcategory
$\mathcal{B}$ is coreflective and invariant under the suspension functor if,
and only if, it is precovering and closed under taking direct summands and
cones. These are extensions of well-known results for AB3 abelian and
triangulated categories, respectively.
  By-side applications of these results allow us: a) To characterize the
coreflective subcategories of a given AB3 abelian category which have a set of
generators and are themselves abelian, abelian exact or module categories; b)
to extend to module categories over arbitrary small preadditive categories a
result of Gabriel and De la Pe\~na stating that all fully exact subcategories
are bireflective; c) to show that, in any Grothendieck category, the direct
limit closure of its subcategory of finitely presented objects is a
coreflective subcategory.
