Example of how to change the event title to 'Venue Booked' if you event is within a 'Private' category.

example.php

<?php //Please do NOT include the opening php tag, except of course if you're starting with a blank file

function tw_ee_EE_Datetime_In_Calendar__to_array_for_json__title( $original_title, $datetime_in_calendar) {
	$event = $datetime_in_calendar->event();

	$categories = $event->get_all_event_categories();

	foreach( $categories as $category ) {
		if( $category->name() == 'Private' ) {
			return 'Venue Booked';
		}
	}

	return $original_title;
}
add_filter( 'FHEE__EE_Datetime_In_Calendar__to_array_for_json__title', 'tw_ee_EE_Datetime_In_Calendar__to_array_for_json__title', 10, 2 );