Endofunctions on a set $X$ form a monoid $End(X)$, which is - as every monoid - a category with a single object $X$.
There seems (to me) to be another conceivable category $E$ of endofunctions on $X$:
Consider as objects of $E$ the elements of $X$. Consider as morphisms of $E$ the ordered pairs $(x,f)$ with $x \in X$ and $f: X \rightarrow X$.
The source of $(x,f)$ is $x$, the target of $(x,f)$ is $fx$.
Composition: $(x,f)\circ(y,g) = (y,f\circ g)$
(if defined, i.e. if $x = gy$)Identities: $\text{id}_x = (x,\text{id})$
Such an "endocategory" $E$ has - among others - the following characteristics;
Each object has the same number of out-arrows ("$|X|^{|X|}$").
There is an obvious equivalence relation between the morphisms: $(x,f) \simeq (x',f')$ iff $f = f'$.
Question 1: Can this equivalence relation be defined categorically ("by dots and arrows") for an endocategory $E$?
Question 2: Where and under which name do endocategories play a role? Are they of any interest?
Question 3: How can endocategories be characterized?