subobject in Set

Question

I cannot compute the coequaliser of $s$ and $id_M$ in $\mathbf{Set}$: My problem is that if $s$ switches $a$ and $b$ in $M=\{a,b\}$ how will then look the coequaliser $M \rightrightarrows M \to U$ where on the l.h.s. is $s$ and $id_M$.I.e. how will look $U$ and the arrow $e$ to it ? Whenever I choose $U=M$ and $e=id_M$ or $e=s$ then neither choice gives when composed the same arrow.