One helpful test to see how "wedded" a preposition is to the verb it follows is to try separating the two in a cleft sentence or a question and see whether the sense of the verb + preposition is retained or lost:
He looked at his cat.
At what did he look? -- His cat.
It was at his cat that he looked.
He looked for his cat.
(?) For what did he look? --His cat.
(?) It was for his cat that he looked.
Though I expect all four of the derived sentences to be dismissed as unnatural, I think that perhaps the native speakers here will agree that the sense of look at is at least preserved in the question and the cleft sentence, and that the sense of look for is either lost or strained to the breaking point in the question and cleft sentence. Indeed, in the cleft sentence It was for his cat that he looked, it almost sounds as if he looked (at something) on behalf of his cat!
Whatever terminology we adopt, it would appear that for is more closely wedded to look in look for than at is to look in look at. That is, grammatically, it would appear that look for forms a tighter verbal unit than look at. Consider, too, that the question Where did he look? could be answered with the prepositional phrase At his cat, but that the same question could not be answered with For his cat. The question needs to be What did he look for?