Custom taxonomy subcategories template page

Yes, this is achievable with the action ‘template_redirect’

I’ve left to write code to find sub-category for yourself!

function my_subcat_template() {

    if ( <is sub category>) {
        include (TEMPLATEPATH . '/category-fun.php');
        exit;
    }
}

add_action('template_redirect', 'my_subcat_template');

Update

This is a better answer for your problem
Put this is functions.php

What it does is, it redefines how wordpress handles templates for categories ( and subcategories)

function new_subcategory_hierarchy() {  
    $category = get_queried_object();

$parent_id = $category->category_parent;

$templates = array();

if ( $parent_id == 0 ) {
    // Use default values from get_category_template()
    $templates[] = "category-{$category->slug}.php";
    $templates[] = "category-{$category->term_id}.php";
    $templates[] = 'category.php';      
} else {
    // Create replacement $templates array
    $parent = get_category( $parent_id );

    // Current first
    $templates[] = "subcategory-{$category->slug}.php";
    $templates[] = "subcategory-{$category->term_id}.php";
    $templates[] = "subcategory.php";

    // Parent second
    $templates[] = "category-{$parent->slug}.php";
    $templates[] = "category-{$parent->term_id}.php";
    $templates[] = 'category.php';  
}
    return locate_template( $templates );
}



add_filter( 'category_template', 'new_subcategory_hierarchy' );

Now you can user subcategory-slug.php or subcategory-id.php for specific subcategories

or subcategory.php in general for all subcategories