I'm reading the paper "Covering Spaces in Representation Theory" of K. Bogartz and P. Gabriel.
Now I'm in section 2, proposition 2.3, on the first three lines concludes that the functor $l \mapsto Hom_{\Lambda}(-,l)$ yields an equivalence $mod \ \Lambda \cong proj \ (ind \ \Lambda)$.
Let $F : mod \ \Lambda \to proj \ (ind \ \Lambda) $ be the above functor. We are going to give a pseudo-inverse, say $G$. If $g: (ind\ \Lambda)^{op} \to Mod \ k$ is an object of $proj \ (ind \ \Lambda)$, we define a functor $G(g): \Lambda^{op} \to Mod \ k$, by $G(g)(X)=g(Hom_\Lambda (-,X))$. It is clear that this functors is well-defined, and then:
$$ G \circ F (X) = G\Big(Hom_{\Lambda}(-,X)\Big): \Lambda^{op} \to Mod\ k $$
is given by $Y \mapsto Hom\Big(Hom_{\Lambda}(-,Y),X\Big)$, and by Yoneda's lemma we get $G \circ F \cong 1_{mod \ \Lambda}$.
For the other composition we have:
$$ F\circ G (X) = Hom_{\Lambda}(-,G(X)) $$
That sends a module $Y : \Lambda^{op} \to Mod \ k$ to the vector space $Hom_\Lambda (Y,G(X))$.
The thing is, I don't see how $$F \circ G \cong 1_{proj (ind \Lambda)} $$
Can anyone help me with this?