Pseudofly
Pseudofly (or pseudo) is an advanced hammerfly technique similar to Speedfly in which the two tees never come into contact, but enables much more dynamic movement through the air when done correctly. With enough practice, a tee (or tees) can pseudofly up, down, and sideways, at will. Pseudofly is generally easier with a dummy, but can also be done with another player as well, although requires great coordination between the two.
Hook timing
The important difference between pseudofly and other Hammerfly is the hook timing. Rather than hooking every two hammer hits, the driver must hook every one 
Hammer hit. The main focus of the driver is to hook just long enough to get into hammer distance, but not too long so that you bump into each other, as that will turn it into a normal hammerfly.
The driver must constantly be paying attention to the distance between the two tees to judge how long to hook for each hammer hit. Once you get comfortable with the hook timing, you will be able to move in all directions at any speed you wish - or even stay in one spot.
Flying down
When pseudoflying down, the driver must hook for shorter time lengths as well as more sparingly. Flying down is basically like falling - except occasionally, the driver hooks and the hammer tee hammers.
When leveling out again, simply increase the rate of hooking and hammering.
Transitioning into speedfly
To transition into speedfly, the driver must be especially careful about the hook timing. When you feel yourself starting to go up, the driver must hook for a little bit longer as speed increases. Once you go fast enough, the driver must hook for every two hammer hits instead of one, like a normal speedfly.
An alternative method to transition to speedfly is to have both tees hammer each other at the same time, and the driver to quickly move left or right, and then level out again. This is much faster method, although the same principles as above still apply.
Use cases
A common use case for pseudofly is to fly through tight horizontal corridors that would be unpractical, or impossible, for any other kind of fly.
Pseudofly is also a common technique used in speedrunning.
Starting a pseudofly
When pseudoflying with a dummy, it's important to control the rate at which the dummy hammers. Therefore, auto-hammer is not enough. You can bind a key to have your dummy hammer at will using the following command in the console (F1):
bind x "+toggle cl_dummy_hammer 1 0"
Once you have that set, you can start pseudoflying.
It is possible to start pseudofly in the air, but it is easiest on flat ground. Start by standing next to the tee, then have the dummy hammer you at a fast rate, hook him at the same time, and start moving in the opposite direction in which you are positioned from the dummy.
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Starting pseudofly from the ground with dummy.
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Starting pseudofly midair with dummy.
When pseudoflying with another player, it's important to communicate with them when it should begin. As the driver, you can do so by hammering, switching to the pistol, and shooting at a fast rate.