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[edit] Katate tori
(also katate mochi, katate dori) One hand holding one hand.
All grabs within Aikido are designed to control the opponent. Katate dori is no exception and it is more than just one hand grabbing another. In order to understand static Katate dori we must look to Katate Dori applied on the move. Consider this, Uke attacks Nage with a Shomen strike of some form, Nage defends with a Shomen block, as the block is applied Uke cuts Nage's arm down into his/her free hand (which grips), simultaneously dropping the hips, applying kokyu and delivering an atemi to the head of Nage (with the now free shomen hand). If proper sword-shape is maintained when the cut-down is done, and the hips are engaged then Nage will be drawn into the strike and will have great difficulty evading it.
When training against a proper Katate dori grab, applied from a situation such as above it is very, very hard for Nage to escape in the split second between the cut down and the grab and strike being delivered. Because of the difficulty of timing this escape, Katate dori is best trained from a static position, where Uke grabs Nage's hand, drops their hips and applies kokyu, as if they were 'frozen' in the process of applying the full control and strike.
If Katate dori is done with this in mind, and Uke attempts to control Nage's centre then Nage will have to apply technique orders of magnitude more 'correctly' than if it were done through a 'lazy' hand grab. This leads to a better training experience for both Nage and Uke and will help improve technique.
As a rule of thumb, when we are Nage, we are often told something along the lines of; "If you're not controlling Uke then you're not doing it right". The same can be said about Uke, when Uke restrains Nage. If Uke is not attempting to control Nage though the restraint, then Uke has not applied the restraint correctly.