Yes, if you have a generic reader and expose the fme_basename, fme_format_short_name and fme_basename, then use a FeatureWriter, in the Writer Parameters set the Output Format to fme_format_short_name and the Base filename to fme_basename (unless you want a different output file name, in which case set it to an attribute containing that name)
In the feature type parameters set the Feature Type Name to fme_feature_type, Geometry from Schema Definition, Check off Dynamic Schema, and set the Schema Sources to your generic reader.
If you would like a different Feature Type Name, then set that to whatever attribute contains the name, but the Schema Definition Name must be set to the fme_feature_type.