I wonder whether following statements holds


If A is an abelian category(or quasi abelian category) having enough projectives, then category of pointed diagram(which means diagram has final object,or for simplicity, one assume the diagram is finite)(A\_D=(D--->A))has enough projectives. 


I want to construct a pair of adjoint functor between this two category. Then use left adjoint of exact functor maps projectives to projectives


Other methods to prove this statement is welcomed



How to construct pair of adjoint functors from category A to category A_D(category of diagrams)
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.


