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If embedding a YouTube video with user-generated captions, you can add ?cc_load_policy=1&cc_lang_pref=en to the end of your YouTube embed URL to display captions on the video by default. This will not work with automatically-generated captions. For more information, see: YouTube Help: Embed videos & playlists.
Vimeo
Vimeo also provides iframe embed code to enable caption display by default. For example, to enable the display of automatically generated english captions, you would add &texttrack=en-x-autogen to your Vimeo embed URL. For more information, see: Vimeo Help Center: Enable captions and subtitles in embeds by default.
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