Here on the page 1 there is a definition of reflexive graph. I need an intuition how it works the morphism $e:X_0\to X_1.$ What is it and to what edge in $X_1$ it sends a vertex from $X_0$?
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$\begingroup$It sends a vertex $v$ to an looped edge $e \circ v$ that is from $v$ to $v$, where $\circ$ is composition of functions. The word "reflexive" in the graph means such edges exist. For details on reflexive graphs, see "Lawvere, Rosebrugh: Sets for mathematics".
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