proving an isomorphism of direct limits

Question

Let $(R,m,k)$ be a local Noetherian ring and $M$ an $R$-module. Let $\left\{I_s\right\}_s$ be a directed system of ideals whose induced topology is equivalent to the $m$-adic topology. Using the definition of direct limit, i proved that $\varinjlim \operatorname{Hom}(R/m^k,M) \cong \varinjlim \operatorname{Hom} (R/I_s,M)$. Is there a proof possibly using exact sequences?