Step 1

Begin with a cylinder. or make your own with the gradient tool.

If you make your own, set the gradient to reflected and linear.

Step 2

Open the object you want to wrap around the cylinder and put it on its
own layer on top.

Step 3

Here is where it all happens:

Filter>Spherize set the amount to 100 and change the mode to horizontal
only.

Click ok

Step 4

See how it looks distorted?

Step 5

Press Ctrl/Cmd+T for free-transform.

Drag the sides in until it matches the size of the shape.

Step 6

This step will add a boost of realism.
Change the top layer to multiply mode. See how it appears to blend into
the cylinder, and it shares the shadows and highlights.

Step 7

If you want a more rounded look on the ends of the cylinder:

Make a huge selection with the circular marquee tool.

Invert the selection. Ctrl/Cmd+Shift+I

Now use the eraser tool to erase the corners away.