Suppose $T:R-Mod \to R-Mod$ is an additive covariant functor that preserves direct limits. (R is commutative, unital. Noetherian if it suits you even). That is, if $(W_{\alpha})_{\alpha \in \Lambda}$ is a direct system of R- modules over a directed set $\Lambda$, then the canonical map
$\varinjlim T(W_{\alpha}) \to T( \varinjlim W_{\alpha}) $ is an isomorphism.
Then why is it true that T also preserves direct sums? By this I mean that if $(M_{\theta})_{\theta \in \Omega} $ is a family of R-modules, then the canonical map
$r : \oplus T(M_{\theta}) \to T(\oplus M_{\theta}) $ is an isomorphism.
My problem: How am I supposed to connect coproducts and direct limits? I know that they're both colimits, but I don't know if a direct sum can be viewed as a direct limit of a system of modules? Since T is additive, it preserves finite direct sums, but that's about all I've got.
Finite direct sums are preserved since we deal with additive functors.
Now use the fact that direct sums are directed colimits of finite direct sums. Explicitly, we have
$$\bigoplus_{i \in I} M_i = \operatorname{colim}_{E \subseteq I \text{ finite}}\, \bigoplus_{i \in E} M_i.$$