The cartridge inside your water filter is meant to be changed regularly. If it's overdue, the filter may be so clogged it's not allowing water to pass through. Unscrew the cartridge chamber door and take out the cartridge. If water falls out, it's a sure sign that water was entering the filter, but debris was keeping it from passing through. Replacement cartridges are available where you bought your filter or from the Culligan website.
Water filters work best in certain conditions. For Culligan filters, the temperature of the water must be 40 to 100 degrees Fahrenheit. In colder climates, tap water may come out colder than 40 degrees. If your tap water is as cold as water in your refrigerator, this could be the problem. If your filter only stops working when you turn on the hot water, the temperature might be too high. Position your faucet temperature control in a way that produces room temperature water, then try your filter again.
Culligan filters only operate when the water pressure is 30 to 100 pounds per square inch. If you have a weak water flow from that faucet, pressure may be insufficient to force water through the filter. Replacing the water lines under your sink with higher-pressure, thicker tubes can boost the flow a bit, but some flow issues require a plumber to diagnose. Don't go above 100 psi, as pressure this strong could force the water through so hard it damages the filter cartridge, leaving contaminants in the water.
Sometimes faulty installation or a damaged part can impede water flow. Remove the filter from the adapter and examine the rubber O-ring that keeps the seal watertight. If it's broken or crumpled, it could block the hole the water flows through to get from the faucet to the filter. Use a new O-ring and screw the filter into the adapter carefully to keep the ring in place. If the O-ring appears fine, inspect the filter for broken or damaged parts. If your filter is still under warranty, it's better to send it to Culligan for service.