Once they've both attempted a hospital corner, . . .

[shootingstar]

(William and Martin at the bedside of their terminally ill friend Colin)
A surge of hope pushes on William's lungs. He's not sure for what, but it propels him to the foot of the bed. 'Let's make sure the sheets are tucked in at the bottom - if his wife sees the state of his toenails, we're rumbled.'
Once they've both attempted a hospital corner, Martin rubs his hands together. 'Right! I'm going to find us both a disgusting cup of tea.'

(From A Terrible Kindness by Jo Browning Wroe, Part IV Midnight Choir, scene 58)

I take "a hospital corner" to mean a "hospital privilege" or a "hospital monopoly" in this context, right?

 

[Skrej]

No, it refers to making the bed and tucking in the bedding corners in a certain manner.

See this guide.

 

[emsr2d2]

When I was a kid, before fitted sheets (the ones with elasticated corners) existed, and my mum was teaching me how to make a bed, she referred to the method of folding the corners of the sheet under the mattress as "hospital corners". I haven't heard the term for at least a decade.