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.

Answer 1

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.