Is the full subcategory of injectives reflective?

Question

Let $mathcal A$ be an abelian category. Denote by $Inj-mathcal A$ the former’s full subcategory consisting of injective objects. Is there any known literature as to when the embedding admits a left adjoint? Or there isn’t any such adjoint?
Thanks in advance.

Zhen Lin · Answer

A reflective full subcategory of an abelian category is closed under kernels and products. Products of injective objects are always injective, but kernels are a problem. Indeed:
Exercise. If an abelian category has enough injective objects and the injective objects form a reflective full subcategory, then every object is injective.
So any abelian category where (a) there are enough injective objects but (b) not every object is injective is a counterexample.

Is the full subcategory of injectives reflective?

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