set one's cap at

• set one's cap for

Probably from French nautical expression ‘mettre le cap sur’, ‘to set a course for’.

set one's cap at

1. (idiomatic) To choose a man as a potential husband (for a girl).
   • 1863, Sheridan Le Fanu, The House by the Churchyard

     To hear her rant, one would have supposed, who had not seen him, that her lank-haired, grimly partner, was the prettiest youth in the county of Dublin, and that all the comely lasses in Chapelizod and the country round were sighing and setting caps at him […]

2. (idiomatic, more generally) To choose something as a goal.
   • Patrick O'Brian, "HMS Surprise".

     How he has escaped marriage until now I cannot tell: the number of caps set in his direction would furnish a warehouse.