Functions can be seen as identifying equivalence classes on a set. For instance, take a set $X$ and an endo-function $f: X \rightarrow X$. Suppose the pre-image of $x_0 \in X$ is a set $A \subset X$, label this preimage with $x_0$. Take this for all elements $x \in X$. Then we have a definition of equivalence classes labelled by elements of $X$. This applies to every function.
Next, consider an endofunctor, $F$, on a category $C$. We take the view of an endofunctor as mapping arrows to arrows. Every arrow has a pre-image and this pre-image is a a set of arrows. Thus, the endofunctor $F$ defines a set of equivalence classes over the arrows of $C$ and they are themselves labelled by arrows of $C$.
Is this a standard interpretation? Does it have a use?