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