existence and uniqueness of projection maping

Question

This is a question on Linear algebra by Greub. Let E be a vector space. Let $C(E)$ be the free vector space formed by E. Let $i_E:E\to C(E)$ be $i_E(x)=f_x$ where $f_x(y)=1$ if $y=x$ and $f_x(y)=0$ otherwise. Show that there is a unique linear mapping $\pi_E:C(E)\to E$ such that $\pi_E\circ i_E=Id$. I feel $\pi_E$ really comes from $i_E$'s fiber. However, I do not see the linearity comes directly.