Evaluation isomorphism for a vertex projective (${\mathrm{Hom}}_{kQ}(kQ{e}_i,X)\cong e_{i}X$)

Evaluation at $e_i$ identifies a module homomorphism $h:kQ{e}_i\to X$ with $h(e_i)\in e_{i}X$. Given $x\in e_{i}X$, the inverse sends $p$ to $px$. The isomorphism is natural and shows that Hom functor evaluation on a vertex projective is exact.