JezK

/**
 * Core class used to implement a Categories widget.
 *
 * @since 2.8.0
 *
 * @see WP_Widget
 */
class WP_Widget_Categories extends WP_Widget {

/**
	 * Sets up a new Categories widget instance.
	 *
	 * @since 2.8.0
	 */
	public function __construct() {
		$widget_ops = array(
			'classname'                   => 'widget_categories',
			'description'                 => __( 'A list or dropdown of categories.' ),
			'customize_selective_refresh' => true,
			'show_instance_in_rest'       => true,
		);
		parent::__construct( 'categories', __( 'Categories' ), $widget_ops );
	}

/**
	 * Outputs the content for the current Categories widget instance.
	 *
	 * @since 2.8.0
	 * @since 4.2.0 Creates a unique HTML ID for the `<select>` element
	 *              if more than one instance is displayed on the page.
	 *
	 * @param array $args     Display arguments including 'before_title', 'after_title',
	 *                        'before_widget', and 'after_widget'.
	 * @param array $instance Settings for the current Categories widget instance.
	 */
	public function widget( $args, $instance ) {
		static $first_dropdown = true;

$default_title = __( 'Categories' );
		$title         = ! empty( $instance['title'] ) ? $instance['title'] : $default_title;

/** This filter is documented in wp-includes/widgets/class-wp-widget-pages.php */
		$title = apply_filters( 'widget_title', $title, $instance, $this->id_base );

$count        = ! empty( $instance['count'] ) ? '1' : '0';
		$hierarchical = ! empty( $instance['hierarchical'] ) ? '1' : '0';
		$dropdown     = ! empty( $instance['dropdown'] ) ? '1' : '0';

echo $args['before_widget'];

if ( $title ) {
			echo $args['before_title'] . $title . $args['after_title'];
		}

$cat_args = array(
			'orderby'      => 'name',
			'show_count'   => $count,
			'hierarchical' => $hierarchical,
		);

if ( $dropdown ) {
			printf( '<form action="%s" method="get">', esc_url( home_url() ) );
			$dropdown_id    = ( $first_dropdown ) ? 'cat' : "{$this->id_base}-dropdown-{$this->number}";
			$first_dropdown = false;

echo '<label class="screen-reader-text" for="' . esc_attr( $dropdown_id ) . '">' . $title . '</label>';

$cat_args['show_option_none'] = __( 'Select Category' );
			$cat_args['id']               = $dropdown_id;

/**
			 * Filters the arguments for the Categories widget drop-down.
			 *
			 * @since 2.8.0
			 * @since 4.9.0 Added the `$instance` parameter.
			 *
			 * @see wp_dropdown_categories()
			 *
			 * @param array $cat_args An array of Categories widget drop-down arguments.
			 * @param array $instance Array of settings for the current widget.
			 */
			wp_dropdown_categories( apply_filters( 'widget_categories_dropdown_args', $cat_args, $instance ) );

echo '</form>';

ob_start();
			?>

<?php
			wp_print_inline_script_tag( wp_remove_surrounding_empty_script_tags( ob_get_clean() ) . "\n//# sourceURL=" . rawurlencode( __METHOD__ ) );
		} else {
			$format = current_theme_supports( 'html5', 'navigation-widgets' ) ? 'html5' : 'xhtml';

/** This filter is documented in wp-includes/widgets/class-wp-nav-menu-widget.php */
			$format = apply_filters( 'navigation_widgets_format', $format );

if ( 'html5' === $format ) {
				// The title may be filtered: Strip out HTML and make sure the aria-label is never empty.
				$title      = trim( strip_tags( $title ) );
				$aria_label = $title ? $title : $default_title;
				echo '<nav aria-label="' . esc_attr( $aria_label ) . '">';
			}
			?>

<ul>
				<?php
				$cat_args['title_li'] = '';

/**
				 * Filters the arguments for the Categories widget.
				 *
				 * @since 2.8.0
				 * @since 4.9.0 Added the `$instance` parameter.
				 *
				 * @param array $cat_args An array of Categories widget options.
				 * @param array $instance Array of settings for the current widget.
				 */
				wp_list_categories( apply_filters( 'widget_categories_args', $cat_args, $instance ) );
				?>
			</ul>

<?php
			if ( 'html5' === $format ) {
				echo '</nav>';
			}
		}

echo $args['after_widget'];
	}

/**
	 * Handles updating settings for the current Categories widget instance.
	 *
	 * @since 2.8.0
	 *
	 * @param array $new_instance New settings for this instance as input by the user via
	 *                            WP_Widget::form().
	 * @param array $old_instance Old settings for this instance.
	 * @return array Updated settings to save.
	 */
	public function update( $new_instance, $old_instance ) {
		$instance                 = $old_instance;
		$instance['title']        = sanitize_text_field( $new_instance['title'] );
		$instance['count']        = ! empty( $new_instance['count'] ) ? 1 : 0;
		$instance['hierarchical'] = ! empty( $new_instance['hierarchical'] ) ? 1 : 0;
		$instance['dropdown']     = ! empty( $new_instance['dropdown'] ) ? 1 : 0;

return $instance;
	}

/**
	 * Outputs the settings form for the Categories widget.
	 *
	 * @since 2.8.0
	 *
	 * @param array $instance Current settings.
	 */
	public function form( $instance ) {
		// Defaults.
		$instance     = wp_parse_args( (array) $instance, array( 'title' => '' ) );
		$count        = isset( $instance['count'] ) ? (bool) $instance['count'] : false;
		$hierarchical = isset( $instance['hierarchical'] ) ? (bool) $instance['hierarchical'] : false;
		$dropdown     = isset( $instance['dropdown'] ) ? (bool) $instance['dropdown'] : false;
		?>
		<p>
			<label for="<?php echo $this->get_field_id( 'title' ); ?>"><?php _e( 'Title:' ); ?></label>
			<input class="widefat" id="<?php echo $this->get_field_id( 'title' ); ?>" name="<?php echo $this->get_field_name( 'title' ); ?>" type="text" value="<?php echo esc_attr( $instance['title'] ); ?>" />
		</p>

<p>
			<input type="checkbox" class="checkbox" id="<?php echo $this->get_field_id( 'dropdown' ); ?>" name="<?php echo $this->get_field_name( 'dropdown' ); ?>"<?php checked( $dropdown ); ?> />
			<label for="<?php echo $this->get_field_id( 'dropdown' ); ?>"><?php _e( 'Display as dropdown' ); ?></label>
			<br />

<input type="checkbox" class="checkbox" id="<?php echo $this->get_field_id( 'count' ); ?>" name="<?php echo $this->get_field_name( 'count' ); ?>"<?php checked( $count ); ?> />
			<label for="<?php echo $this->get_field_id( 'count' ); ?>"><?php _e( 'Show post counts' ); ?></label>
			<br />

<input type="checkbox" class="checkbox" id="<?php echo $this->get_field_id( 'hierarchical' ); ?>" name="<?php echo $this->get_field_name( 'hierarchical' ); ?>"<?php checked( $hierarchical ); ?> />
			<label for="<?php echo $this->get_field_id( 'hierarchical' ); ?>"><?php _e( 'Show hierarchy' ); ?></label>
		</p>
		<?php
	}
}

JezK

Edit File: class-wp-widget-categories.php