Verb for librarian giving book to library visitor

Issue
You borrow a book at the library. The librarian issues a book.

2 issue
transitive verb
2a : to put forth or distribute usually officially
Examples of ISSUE
Each employee will be issued an identification card.
The Post Office will issue a new first-class stamp.
The company plans to raise money by issuing more stock.
The bank will be issuing a new credit card.
the bank's newly issued credit card.
The king issued a decree forbidding all protests.
A severe storm warning has been issued.
The police have issued a warrant for her arrest.

[Edit-1: Per @Mustafa & @sidewaysmilk]
Lend is the usual alternative, esp. in the US.

Library / librarian may lend and/or issue books.

lend: to permit the use of (something) with the expectation of return of the same or an equivalent.

issue: the act of sending or giving out something; supply; delivery

In libraries where I've worked in the UK, the librarian issues books.

From my decade of experience as a librarian, I would say that in most libraries, the librarian does not check out books. That job is usually done by a library clerk... But maybe I'm being facetious.

In the US, we often use 'loan'. The books is on loan. Request an inter-library loan... The library loaned the book to the patron.

Anyway, the librarian (or whoever) does not loan the book. The librarian doesn't own the book. The Library loan/lends the book. The librarian checks the book out.

Or 'The library clerk scanned the barcode on the inside front cover of the book, tore off the return slip from the receipt printer, demagnified the security strip and slid the books back across the counter.' Because, you know, that's actually what happens.