[{"name":"Some name","products":[{"product_name":"a"},{"product_name":"b"},{"product_name":"c"}]},{"name":"Some name","products":[{"product_name":"c"},{"product_name":"b"},{"product_name":"a"}]},{"name":"Some name","products":[{"product_name":"c"},{"product_name":"a"}]},{"name":"Some name","products":[{"product_name":"c"},{"product_name":"b"},{"product_name":"a"},{"product_name":"d"}]},]

db.collection.update({"name":"Some name"},[{$set:{products:{$cond:[{$setEquals:[["a","c"],"$products.product_name"]},"$products",[{"product_name":"a"},{"product_name":"c"}]]}}}],{multi:true})