source: wiles/1.1.1/inc/widgets/widget-category.php

<?php
add_action( 'widgets_init', 'wiles_category_posts_widget' );
function wiles_category_posts_widget() {
        register_widget( 'wiles_category_posts' );
}
class wiles_category_posts extends WP_Widget {

        function wiles_category_posts() {
            $txt = sprintf(__(' - Category Posts', 'wiles'));
                $widget_ops = array( 'classname' => 'wiles-category-posts' );
                $control_ops = array( 'width' => 250, 'height' => 350, 'id_base' => 'wiles-category-posts-widget' );
                $this->WP_Widget( 'wiles-category-posts-widget',theme_name . $txt, $widget_ops, $control_ops );
        }
        
        function widget( $args, $instance ) {
                extract( $args );

                
               $title = apply_filters('widget_title', $instance['cat_posts_title'] ); 
               $no_of_posts = $instance['no_of_posts']; 
        $cats_id = $instance['cats_id'];
                
                    echo $before_widget;
                        if ( $title){
                    echo $before_title;
                        echo $title ; 
                    echo $after_title;} ?>
                                <ul>
                                        <?php wiles_last_posts_cat($no_of_posts , $cats_id)?>   
                                </ul>
                <div class="clear"></div>
        <?php 
                echo $after_widget;
        }

        function update( $new_instance, $old_instance ) {
                $instance = $old_instance;
                $instance['cat_posts_title'] = strip_tags( $new_instance['cat_posts_title'] );
                $instance['no_of_posts'] = strip_tags( $new_instance['no_of_posts'] );
                $instance['cats_id'] = implode(',' , $new_instance['cats_id']  );
                
                return $instance;
        }

        function form( $instance ) {
                $defaults = array( 'cat_posts_title' =>__( 'Category Posts' , 'wiles'), 'no_of_posts' => '5' , 'cats_id' => '1' );
                $instance = wp_parse_args( (array) $instance, $defaults );
                
                $categories_obj = get_categories();
                $categories = array();

                foreach ($categories_obj as $pn_cat) {
                        $categories[$pn_cat->cat_ID] = $pn_cat->cat_name;
                }
                ?>
                <p>
                        <label for="<?php echo $this->get_field_id( 'cat_posts_title' ); ?>"><?php _e('Title:', 'wiles'); ?> </label>
                        <input id="<?php echo $this->get_field_id( 'cat_posts_title' ); ?>" name="<?php echo $this->get_field_name( 'cat_posts_title' ); ?>" value="<?php echo $instance['cat_posts_title']; ?>" class="widefat" type="text" />
                </p>
                <p>
                        <label for="<?php echo $this->get_field_id( 'no_of_posts' ); ?>"><?php _e('Number of posts to show:', 'wiles'); ?> </label>
                        <input id="<?php echo $this->get_field_id( 'no_of_posts' ); ?>" name="<?php echo $this->get_field_name( 'no_of_posts' ); ?>" value="<?php echo $instance['no_of_posts']; ?>" type="text" size="3" />
                </p>
                <p>
                        <?php $cats_id = explode ( ',' , $instance['cats_id'] ) ; ?>
                        <label for="<?php echo $this->get_field_id( 'cats_id' ); ?>"><?php _e('Category:', 'wiles'); ?> </label>
                        <select multiple="multiple" id="<?php echo $this->get_field_id( 'cats_id' ); ?>[]" name="<?php echo $this->get_field_name( 'cats_id' ); ?>[]">
                                <?php foreach ($categories as $key => $option) { ?>
                                <option value="<?php echo $key ?>" <?php if ( in_array( $key , $cats_id ) ) { echo ' selected="selected"' ; } ?>><?php echo $option; ?></option>
                                <?php } ?>
                        </select>
                </p>
        <?php
        }
}
?>