Look use this simple example.

100mm f 2 @ 20ft has 1.1ft dof.

Put a 2xTC on above for a 200mm f5.6 @ 20ft and you get 0.54ft dof.

The focal length increased but the actual aperture size (not the f number) stayed the same, so you get less dof.

Sorry, but a 100mm f/2 lens with a 2x TC would be a 200mm f/4 lens, not f/5.6.

So, let's repeat your example with correct math (and a specified sensor size, since that does matter):

100mm f/2 on FF at 20 feet, DoF = 1.54 feet

200mm f/4 on FF at 20 feet, DoF = 0.75 feet

Now, back up so the shot with the 2x TC has the same framing as the shot without the TC:

200mm f/4 on FF at

**40** feet, DoF = 3.07 feet

Plug those numbers (100/2, 200/4, 20 ft vs. 40 ft) into your DoF calculator of choice...you may get slightly different absolute values (different calculators make different assumptions about CoC, print size and viewing distance, which affects the calculated DoF values), but the relative differences will be the same.

Compared to a bare lens:

- Adding a TC and shooting at the same distance means shallower DoF (and a smaller FoV
- Adding a TC and moving back so the FoV is identical means
**deeper** DoF