# universal property of the direct colimit

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 $v\in F(\lambda),\; \lambda\le \lambda '$.

You can endow $\operatorname{colim}\limits_{\lambda} F$ with an $R$-module structure as follows: For $\lambda_0\in\Lambda$, $s_{\lambda_0}:F(\lambda_0)\to \operatorname{colim}\limits_{\lambda} F$ which is the composition of the inclusion into the coproduct and the projection, let

$$r\cdot s_\lambda (v)=s_\lambda (rv)$$ and $$s_\lambda (v)+ s_{\lambda '} (v')=s_\mu (F(i_{\lambda,\mu})(v)+F(i_{\lambda ',\mu})(v'))$$ for $\lambda, \lambda '\le\mu$.

My question is: What is the universal property of the direct colimit?

The universal property of colimit doesn't depend on a category in which it is calculated. In the case of the category of $R$-modules the universal property means that for any $R$-module $M$ and family of homomorphisms of $R$-modules $f_{\lambda}\colon F(\lambda)\to M$, such that for any morphism $i_{\lambda,\lambda'}$ of $(\Lambda,\le)$ the equality $f_{\lambda'}\circ F(i_{\lambda,\lambda'})=f_{\lambda}$ holds, there exists a unique homomorphism of $R$-modules $f\colon\text{colim}_{\lambda}F\to M$, such that $f\circ s_{\lambda}=f_{\lambda}$ for every $\lambda\in\Lambda$.