To write or present someone's biography or life story in the detailed, conversational style characteristic of James Boswell's work.
From Boswell plus the suffix '-ize' (meaning to make, practice, or treat in the manner of). This converts the biographical style into a verb describing the action of writing biography that way.
The verb 'to boswellize' is a perfect example of how influential people's methods become—verbing a person's name to mean 'do what they do' shows their approach fundamentally changed how we think about biography.
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