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"he's eager" is correct and usable in written English.
It is an idiom which is typically used to indicate that someone is enthusiastic, excited and keen to do something. For example, "He's eager to get started on the project."
Exact(58)
Quarterback Matt Barkley acknowledged after Sunday's 20-17 loss to the Lions at Ford Field that he's eager for Jeffery's return.
A bold deal shows that Yang isn't afraid of losing Yahoo's independence, rather, he's eager to use it.
He's eager for Don's approval but quickly frustrated when Don outshines him once again.
But he's eager to discuss his wife's new novel, for which he served as something of a muse.
Robert Pattinson plays Pearce's roadmate, a simple-minded American called Rey, and he's eager, but anxious in the role.
He's eager to get into the studio to record the band's next full-length album with all-new material.
"He's eager.
He's eager to learn.
He's eager to branch out.
But he's eager to ice her.
But now he's eager to leave.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com