I am currently working through "Generic figures and their glueings" by Marie La Palme Reyes, Gonzalo E. Reyes and Houman Zolfaghari. Exercise 1.2.1.3 asks the following:
Let $\Delta_1$ be the monoid of the endomorphisms of the arrow $A$ of reflexive graphs. As a category, $\Delta_1$ has one object and three arrows: $1$, $\delta_0$ and $\delta_1$ with relation $\delta_i\delta_j = \delta_i$ (where $\delta_0$ is the constant map that sends the whole arrow in its source and $\delta_1$ the arrow in its target.) Show that $\mathsf{Set}^{\Delta_1^{op}}$ may be identified with the category of reflexive graphs.
The category of reflexive graphs is defined to be the category $\mathsf{Set}^{\mathbb{C}^{op}}$ where $\mathbb{C}$ is the category generated by $V \xrightarrow{s} A$, $V \xrightarrow{t} A$, $A \xrightarrow{l} V$.
As I understand it, the exercise requires to show the equivalence of the categories $\mathsf{Set}^{\Delta_1^{op}}$ and $\mathsf{Set}^{\mathbb{C}^{op}}$.
My question: Is there a typo in the exercise? Shouldn't it be $\delta_i\delta_j = \delta_j$ instead of $\delta_i\delta_j = \delta_i$?
The equivalence of these categories (after fixing the possible typo) is also mentioned in section 2 of reflexive graph in ncatlab.