In a directed system $(M_i)_{i\in I}$ of $R$-modules ($R$ a commutative ring) for a directed poset $I$ there is in particular for any $i$, $j$ a module $M_k$ with linear maps $f_{ki}:M_i\to M_k$ and $g_{kj}:M_j\to M_k$. This construction is not unique, i.e. there might be a module $M_{k'}$ with linear maps $f_{k'i}:M_i\to M_{k'}$ and $g_{k'j}:M_j\to M_{k'}$.
My question is: Are these upper bounds $M_k$ and $M_{k'}$ comparable in the sense that there is a map $g:M_k\to M_{k'}$ commuting with the $f,f'$ and $g,g'$ in a suitable sense? If so, how does this follow from the axiom of a directed system? If not, would this hold for filtered colimits?
In a directed system, since there can be at most one map $f_{ji}:M_i\to M_j$ for each $i,j$, we just need the condition $f_{ki}=f_{kj}\circ f_{ji}$.
Added: there may be two maps $f_{ki}$ and $f_{k'i}$, but we can find $M_l$ and $f_{lk},f_{lk'}$ and we automoatically have $f_{lk}\circ f_{ki}=f_{lk'}\circ f_{k'i}$. This follows from the axiom that any two element in a directed set has an upper bound.
But for a filtered system, if there are two maps $f_{ji},f'_{ji}:M_i:\to M_j$ for some $i,j$, we ought to have a map $g_{kj}:M_j\to M_k$ such that $g_{kj}\circ f_{ji}=g_{kj}\circ f'_{ji}$