"fasten on" is a correct and usable phrase in written English.
It means to focus or pay attention to something. Example: The detective's eyes seemed to fasten on the suspect's nervous gestures.
And: 'What Don Juan realises in action is an ethic of quantity, whereas the saint, on the contrary, tends towards quality.' He carried the philosophy further, claiming that a mother or an uxorious wife necessarily had 'an enclosed heart' because it is 'turned away from the world' to fasten on one object.
"Movie audiences fasten on to one aspect of the actor, and then they decide what they want you to be," Mr. Widmark once said.
The crocodiles moved in straight away - we watched one fasten on to the haunch of a wildebeest which nevertheless struggled its way, bellowing its head off, across the river.
The concrete meaning – the one that, until now, Raine would have had us fasten on to – is that the torch is there to allow the fishermen to see the fish.
We fasten on the principal feature of the action.
If it isn't fastened on tight enough, make the slits deeper.
Something was tugging at my foot: I turned around to see her yellow teeth fastened on one of my flippers.
Hereafter, notices were to be fastened on the locker-room bulletin board with four thumbtacks, one at each corner.
For a while they fastened on Kissinger's facial expressions.
I fastened on a few coincidental biographical similarities.
