Question

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

Answer 1

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.

Answer 2

There are several pairs of adjoint functors of the kind you desire but it isn't clear to me if any (or all) of them will give you enough projectives in $A^D$.

For example for each $d$ in $D$ $\iota_d$ which sends an object of $A$ to the diagram that is $A$ at $d$ and zero elsewhere and does the obvious thing on morphisms is left adjoint to the functor $\pi_d$ that sends a diagram to its value at $d$.

Also if $D$ is pointed and $A$ has $D$-colimits then $\mathrm{colim}\colon A^D\rightarrow A$ is left-adjoint to the constant functor that sends $X$ in $A$ to the diagram that is $X$ everywhere and all morphisms are $\mathrm{id}_X$ (see Wikipedia).

These left adjoints will all map projectives to projectives.