# definition of the directed colimit of a functor

Let $(\Lambda,\le )$ be a directed set, which we can understand as a small category: The set of all objects is $\Lambda$ and for $\lambda,\lambda '\in \Lambda$ there exists an unique morphism $i_{\lambda,\lambda '}:\lambda\to \lambda '$ iff $\lambda\le \lambda '$. Otherwise we don't have any morphisms.

Let $R-$MOD be the category of $R$-modules, $F:(\Lambda,\le )\to R$-MOD a functor. The directed colimit of $F$ is

$$\operatorname{colim}\limits_{\lambda} F:=\coprod\limits_{\lambda\in \Lambda}F(\lambda)/\sim ,$$where $v\sim F( i_{\lambda,\lambda '})(v)$ for $v\in F(\lambda),\; \lambda\le \lambda '$.

My question: is $\sim$ really an equivalence relation? This realtion isn't symmetric, or am I wrong?

Best

EDIT : [I will assume that by $\coprod\limits_{\lambda\in \Lambda}F(\lambda)$ you mean $\bigoplus\limits_{\lambda\in \Lambda}F(\lambda)$ using the general notation for coproduct (and not the disjoint union, which would be wrong).] Actually for directed colimits it doesn't matter.
Then you have to quotient by the equivalence relation that is generated by $v\sim F( i_{\lambda,\lambda '})(v)$ (which you have to show is a congruence).