ct.category theory - Category = Groupoid x Poset?

Given any locally small category, $C$, the collection of all isomorphisms forms a subgroupoid, $G subseteq C$, where $Ob(G) = Ob(C)$ and $Hom_G(A,B) = left ( f in Hom_C(A,B) : exists g, h in Hom_C(B,A) g circ f = id_A, f circ h = id_B right ) $.

Because $G$ is a groupoid, it determines an equivalence relation, $R$ on the objects and morphisms of $C$ such for $A, B in Ob(C)$:

$A equiv_R B Longleftrightarrow Hom_G(A,B) neq emptyset$

And for $f, g in Hom_C(A,B)$:

$f equiv_{R_{A,B}} g Longleftrightarrow exists h_B in Hom_G(B,B), h_A in Hom_G(A,A) : h_B circ f = g circ h_A$

If I understand what you are asking, then the quotient $C/R$ should be the 'poset' you want.