adverb
tastefully; respectfully; decently
Describes doing something in a way that maintains appearances, shows proper form, or is socially presentable. Often used when one wants to avoid embarrassment or keep up a good front.
体裁よく断るのは難しい。
It's difficult to decline politely while saving face.
彼は体裁よくその場を収めた。
He handled the situation decently, keeping up appearances.
体裁 is the noun meaning 'appearance' or 'form'; 体裁よく is the adverbial form meaning 'in a presentable manner'.
Adverbial form of the noun 体裁 (appearance, form) + よく (adverbial form of よい).