my understanding of these is that they are similar to the secondary trigger on some rifles. Night latches on trap dogs are a secondary notch filed slightly "lower" and on the very end of the dog itself. To set or activate the night latch then, you place the dog on the higher or farther back level from the dog tip. Then, by slightly pulling down on the pan trigger, the point where the pan is contacting with the dog will slide down toward the tip of the dog farther until it slips down into this lower stepped down filed section on the end of the dog. You should here a "click" when the pan area engages in this lower night latch part of the dog. This night latch set position is the final firing position of the trap. The reason it is called a night latch is that you can set this trap, pull down on the pan until you hear that second "click", and know that it is set without having to "see" the assembly.
The top and the tip of the dog, seen from the side, should look like this:
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Pretty rough drawing, but that is the idea. That the rear-most top portion of end of the dog is the initial first setting position, the secondary notched area is the "night-latch" set position. When the pan area engages down into that filed notch, you hear the "click".
Jim-NE