Taking direct limits is an exact functor in the category of modules. It has been discussed extensively here.
What I ask is: I know that taking direct limits is not an exact functor in other categories. Our professor mentioned it when discussing Cech cohomology of sheaves, describing the n-th Cech cohomology group as a direct limit. (I can provide details if it's necessary)
Can you please help me find an example in some category where taking direct limits is not an exact functor? Categories of groups or topological spaces maybe?