I'm a bit bothered by something I've come across, and I'd like to know if the misunderstanding is my own (likely) or the author's (unlikely).
Let $\mathcal{C}$ be a category. An arrow $e : A \to A$ in $\mathcal{C}$ is idempotent if $e \circ e=e$. If $\mathcal{E}$ is a class of (not necessarily all) idempotents in $\mathcal{C}$ then we can form a category $\mathcal{C}[\check{\mathcal{E}}]$ whose objects are the members of $\mathcal{E}$ and whose arrows $f : (e:A \to A) \to (d:B \to B)$ are arrows $f : A \to B$ in $\mathcal{C}$ satisfying $d \circ f \circ e=f$.
Something confuses me here. Say $1_A, 1_B \in \mathcal{E}$ and $(e,d) \ne (1_A,1_B)$. We have $d\circ f \circ e=f=1_B \circ f \circ 1_A$. So in the definition of $\mathcal{C}[\check{\mathcal{E}}]$ given above, $f$ is both an arrow $e \to d$ and an arrow $1_A \to 1_B$.
To me this seems absurd. But is it? Can we declare $f : e \to d$ and $f : 1_A \to 1_B$ to be distinct arrows in $\mathcal{C}[\check{\mathcal{E}}]$ despite having the same underlying arrow in $\mathcal{C}$?