Ok then, I actually was getting the subcats with the wrong method.
in $parent_cat = get_cat_ID( $cat_name );
$cat_name
wasn’t a name but un slug. First mistake.
So I corrected with $parent_cat = get_category_by_slug( $cat_slug );
The final code is :
// Get the parent cat thanks to the page's cat slug
$parent_cat = get_category_by_slug( $cat[1] );
$args = array(
'type' => 'post',
'parent' => $parent_cat->term_id,
'order' => 'DESC'
);
// child cats
$categories = get_categories( $args );
// loop on the child cats to get the sub cats object
foreach ($categories as $key => $value) {
return $value;
}
That’s it.