To make something suitable for display, or to turn something into an exhibition.
From exhibition + -ize (suffix creating verbs meaning 'to cause to be' or 'to perform an action'). The -ize suffix is highly productive in English for creating verbs.
This is a very rare word—most museums and galleries simply say 'to exhibit' or 'to display' rather than 'exhibitionize,' showing that even productive word-formation patterns don't guarantee widespread adoption.
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