Projective objects of easy functor category. Let $I$ be the category with two objects ${1,0}$ and only one "nontrivial" arrow from $1$ to $0$. Let $A$ be an abelian category (the category I want to apply this is a category of modules over a k-algebra). Let $C:=[I,A]$ the category of functors from $I$ to $A$. In simple terms an object of C is a morphism in $A$ and a morphism in $C$ is a commutative diagram in $A$. Is there any characterization of projective objects of $C$? Is it true that $F \in C$ is projective if and only if $F(0)$ and $F(1)$ are projectives? Otherwise is we call $f$ the morphism from $F(1)$ to $F(0)$ is it true that $F$ is projective if and only if $coker(f)$ is projective? Even if this questions seems very easy I don't have an unswer but I really need it. I would be grateful to anybody who could answer.
 A: I think the projective objects are precisely the direct-sum-inclusions
$\sigma_w:P\to P\oplus Q$ with $P$ and $Q$ projective in $A$.
It is easy to see that such objects are indeed projective in $[I,A]$.
Conversely assume that $f:U\to V$ is projective in $[I,A]$. Given an epimorphism $e:Y\twoheadrightarrow V$ we can form the pullback
$\require{AMScd}$
\begin{CD}
X @> f'>> Y\\
@V e' V V @VV e V\\
U @>>f> V\,\,.
\end{CD}
Since epis are stable under pullbacks in abelian categories, $e'$ is epic, thus the pair $(e',e)$ is an epi from $f'$ to $f$ in $[I,A]$. By projectivity assumption on $f$ we obtain a section which in particular gives a section of $e$. Thus, $V$ is projective in $A$.
Now consider the square
$\require{AMScd}$
\begin{CD}
U @>\sigma_1>> U\oplus V\\
@V 1 V V @VV [f,1] V\\
U @>>f> V
\end{CD}
Again by projectivity assumption on $f$ we get a section $s$ of $[f,1]$ satisfying $sf=\sigma_1$. Let $\pi_1:U\oplus V\to U$ be the first projection.
We have $\pi_1\circ s\circ f=\pi_1\circ\sigma_1=1$. Thus, $f$ has a retraction $\pi_1\circ s$, and therefore is a direct-summand-inclusion, i.e.
$(U\stackrel{f}{\to}V)=(U\stackrel{\sigma_1}{\to}U\oplus W)$ for some $W$. Finally, all direct summands of projective objects are projective.
