Let $I$ be a cofiltered index category, $A_i$ be a directed system of $R$-modules. Is the following true?
$$\mathrm{Hom}(\lim_IA_i,B)=\mathrm{colim}_I(\mathrm{Hom}(A_i,B))$$
(Note this is not true for arbitrary limits, for example if $I$ is the discrete category.)
No. For instance, let $R=\mathbb{Q}$, $I=\mathbb{N}$, and $A_n=\mathbb{Q}^n$ connected by the obvious projection maps. The inverse limit $\lim A_n$ is then $\mathbb{Q}^\mathbb{N}$, which has uncountable dimension. In particular, $\operatorname{Hom}(\mathbb{Q}^\mathbb{N},\mathbb{Q})$ is uncountable. But $\operatorname{Hom}(\mathbb{Q}^n,\mathbb{Q})$ is countable for each $n$, and so it follows that the direct limit is also countable.
In fact, you can find a counterexample for any nonzero module $B$ at all: see my answer here.