If D is small and A has enough projectives and has infinite sums then $A^D$ has enough projectives. For the proof, see Weibel, "An introduction to homological algebra", 2.3.13 on p.43. The proof is a version of Godement's argument that the category of sheaves of abelian groups has enough injectives.

If D is small and A has enough projectives and has infinite sums then $A^D$ has enough projectives. For the proof, see Weibel, "An introduction to homological algebra", 2.3.13 on p.43. It contains the adjoint you are apparently looking for.

The proof is a version of Godement's argument that the category of sheaves of abelian groups has enough injectives.

If D is small and A has enough projectives and has infinite products then $A^D$ has enough projectives. For the proof, see Weibel, "An introduction to homological algebra", 2.3.13 on p.43. The proof is a version of Godement's argument that the category of sheaves of abelian groups has enough injectives.

If D is small and A has enough projectives and has infinite sums then $A^D$ has enough projectives. For the proof, see Weibel, "An introduction to homological algebra", 2.3.13 on p.43. The proof is a version of Godement's argument that the category of sheaves of abelian groups has enough injectives.