I'm reading the proof of theorem $2.15$ of Quiver Representations by Karl Schiffler. The author states the following:
Let $Q$ be a finite acyclic quiver, $M=(\left\{M_i\right\}_{i\in Q_0}, \left\{\phi_{\alpha}\right\}_{\alpha\in Q_1})$ a quiver and let $d_i=\dim M_i$. We define a projective representation $P_0:=\bigoplus_{i\in Q_{i}}d_iP(i)$, here $P(i)$ denotes the indecomposable projective representation associated to the vertex $i$. Recall that $P(i)_j$ is the vector space with basis given by all paths from $i$ to $j$.
The author then states that the set $$\left\{c_{ij}\mid i\in Q_0, c_i \mbox{ is a path such that $s(c_i)=i$ (i.e. $c_i$ begins in $i$), j=1, \dots , d_i}\right\}$$ is a basis of $P_0$. However, I do not understand why this is so.
When I read the basis in words, I read that for each vertex $i$ we consider $d_i$ times a path $c_i$ that starts in $i$.
Now I believe this is incorrect, as $P(i)_j$ is spanned by all paths from $i$ to $j$, $P(i)$ is spanned by all paths beginning at $i$. Hence a basis of $P_0$ is given by
$$\bigcup_{i\in Q_0}\left\{\mbox{$d_i$ times all paths beginning at $i$}\right\}.$$
Am I wrong or is the author just using some (mild) abuse of notation?