What do you mean by ‘we can update’ - we who?
The Database is not an ‘active entity’ - it cannot do stuff on it’s own - it only responds to queries. So ‘it’ cannot update the cache.
Why not always read from the caching service?
The caching service may be designed to return the shortened URL if present in the cache. If not present in the cache, the caching service may read from the DB and update the cache and then return the entry (updating the cache may be performed lazily)
The other way is that the application service may be entrusted with the responsibility of updating the cache when there’s a miss!