Definition Let $A$ and $B$ be any objects in category $C$. We say that $A$ is retract of $B$, notated by $A\le_r B $, iff $\exists s:A\rightarrow B, r:B\rightarrow A$ such that $r\circ s=1_A$.
It is easy to see that $\le_r$ is reflexive and transitive. But, I am having difficulty seeing that $\le_r$ is also anti-symmetric. In other words, is following statement true:
$$[(A\le_r B)\wedge (B\le_r A)]\Rightarrow (A\cong B).$$
Motivation behind this question stems from concept of "least co-separator". An example of least co-separator is 2-set in category of sets and functions. One may define a least co-separator as a co-separator A such that $\forall co-separator B\in C, A\le_r B$. From this, given that $\le_r$ is anti-symmetric, uniqueness of co-separator (upto isomorphism) follows. I would much appreciate any input on this too.
If the statement about anti-symmetry is false, could you please provide any elementary example, if possible?