Before beginning, turn off the water at the shutoff valves under the sink (Image 1). Open the faucet to drain any excess water.
Use a small adjustable wrench to disconnect the hot- and cold-water supply lines from the shutoff valves (Image 2).
Using a basin wrench, reach up behind the faucet, and unscrew the coupling nuts connecting the supply tubes to the faucet (figure 1).
Use the basin wrench to remove the mounting nuts holding the faucet in place (figure 2).
With the mounting nuts disconnected, lift out the old faucet, and use an adjustable wrench to disconnect the sprayer hose from the assembly (figure 3).
Lift out the sprayer hose. After the faucet is removed, use a scouring pad to clean the sink surface thoroughly. If the sprayer-hose escutcheon cap is in good shape, you may want to leave it on. If not, replace it with the one included with the new faucet.
Slip the new sprayer hose down through the sprayer hole, and feed it up through the center faucet hole. Use an adjustable wrench to connect the hose to the sprayer nipple.
It's time to connect the new faucet's fittings with the water-supply lines. The manufacturer's instructions may direct you to set the new faucet assembly in place, then crawl back under the sink to make the new connections. Save yourself from struggling in cramped quarters by making all the connections you can before setting the new assembly in place. Then feed the connected lines down through the hole in the sink. With the attachments at the base of the faucet assembly already in place, the only work you'll need to do under the sink is to connect the supply lines to the shutoff valves.
The hot and cold supply tubes on the faucet may vary in length, depending on the brand. You may want to attach flexible connectors to adapt the faucet for a particular installation. If so, attach them to the fittings on the faucet's supply tubes. Use two wrenches to tighten each connection -- one to hold the faucet-fitting stationary and one to turn and tighten the female fitting on the flexible connector. This two-wrench technique will help prevent twisting and damaging the faucet's copper supply lines.
(Note: Depending on the size of the hole, you may have a tight fit and thus be able to attach only one flexible connector before seating the faucet assembly. In that case, you may need to wait and attach one of them from underneath the sink once the faucet's in place.)
Insert and feed the connectors and supply lines into the center hole, and, making sure that the faucet's gasket lines up correctly, seat the faucet (Image 1).
From under the sink, install a washer and nut -- or a plastic nut-washer combo (Image 2) -- to the end bolts that will hold the faucet in place. Tighten the nuts by hand, then tighten with a basin wrench.
Attach the second flexible connector if you haven't already (Image 1), and attach the male ends of the connectors to the shutoff valves (Image 2). Tighten the connections with an adjustable wrench. Turn the water on at the shutoff valves, and turn on the faucet to check whether it's working properly.