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The part of the sentence "mutual admiration between" is correct and usable in written English.
You can use it whenever you are expressing the mutual admiration that exists between two or more individuals. For example, "The new couple displayed a mutual admiration between them that was obvious to all their friends."
Exact(11)
Corporate success does not necessarily mean mutual admiration between worker and management.
Strauss heartily agreed, and an artistic partnership was born in which mutual admiration between composer and writer ran deep.
With another conductor, it's often "a display of mutual admiration between him and God," said Eugene Izotov, the principal oboist of the Chicago Symphony, who played Mozart's Oboe Concerto with Mr. Morlot.
It is just one chapter in a long history of mutual admiration between French and Russian artists, a sort of alliance on the flanks of the Teutonic behemoth at mid-Continent.
There is mutual admiration between the administration and unions at Western Connecticut State University in Danbury, and at UConn's Storrs campus, there has been a long tradition of the two sides getting along.
Much publicity was given to some of his earlier, off-the-pitch mishaps – including a jacuzzi tryst at Bayern's training grounds – but not as much as has been written about the mutual admiration between him and Merkel.
Similar(49)
Although he claims to have been a hit with the ladies from the get-go ("It's always been a mutual admiration society between me and the girls," he tells me soon after we sit down in the bus's grim little kitchen area), one suspects that his early successes had more to do with charm than looks.
It's the look of mutual admiration or pain between two warring soldiers, the friendship that springs between children of hostile races, the love that grows between a couple so different they initially can't stand each other.
There's clearly a lot of mutual admiration and joshing between them.
Within the quartet there was mutual admiration and, at least between Olivier and Gielgud, an edge of rivalry.
A bit of a mutual admiration society has developed between the two finalists in Qualcomm's $10 million Tricorder XPRIZE competition.
More suggestions(15)
mutual love between
mutual embrace between
mutual mistrust between
mutual estrangement between
mutual dependency between
mutual interest between
mutual deference between
mutual influence between
mutual confidence between
mutual distrust between
mutual understanding between
mutual society between
mutual one between
mutual loathing between
mutual cooperation between
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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