Illustrator Tutorial on Mapping the Artwork to 3D Object

First, create a label. Drag it to the Symbols library.

Create the 3D object using the Pen tool and draw half of the shape. Use the Direct Selection tool and choose the two end points. Apply Path>Average>Vertical to the two end points to make them even.

Change the Stroke to Fill. With the Direct Selection tool choose 3D > Revolve.

Check Preview and choose Right Edge.

Now we have an object for mapping. Don't leave the 3D dialog box.

Choose Map Art.

Select the new symbol. Use the Surface arrows to get to the right section. Watch the actual object as you go through the choices.

Make alterations to the placement.

If you are unsatisfied by the placement of the object, then press Control+Z and start over. After that, use the Transparency Flattener to complete the project. Group the piece and Save. Include it in the Symbols palette.