class Base
{
fn_base(){...};
};

template < class T > 
class Derived : public Base
{
fn_derived(){...};
T value_;
};

I have 2 classes as you can see above.
Additionally I have a map that cointains shared_ptr<Base> + a specific id.

std::map< const std::string id, std::shared_ptr<Base>> objects_;

Now I want to store an object of type Derived<T> in the map ( works fine ) and receive the value of it somewhere else ( here is the problem )

template < class T >
T getObjectValue( const std::string id ){
    if ( object_.find(id) != objects_end()){
        return std::dynamic_pointer_cast<Derived<T>>(objects_.at(id))->getValue();
    }
}

This :

return std::dynamic_pointer_cast<Derived<T>>(objects_.at(id))->getValue();

Doesn't work. But when I replace T with a specified type it works ?

return std::dynamic_pointer_cast<Derived<int>>(objects_.at(id))->getValue();

Can someone explain that ? And if you can, would you mind giving me a hint on how I can work around that ?

Now I want to store an object of type Derived<T> in the map ( works fine )[...]

I meant a std::shared_ptr<Derived<T>>

Please show us a reproducible example, don't merely paste extracts that don't even compile.
My guess (as you didn't show more code, which would be required for a proper assessment) is that the actual code - not the one you just typed into the post - contains a template-id for getValue :

template < class T >
T getObjectValue( const std::string id ){
    if ( object_.find(id) != objects_end()){
        return std::dynamic_pointer_cast<Derived<T>>(objects_.at(id))->getValue<...>();
    }
}

Instead, try

template < class T >
T getObjectValue( const std::string id ){
    if ( object_.find(id) != objects_end()){
        return std::dynamic_pointer_cast<Derived<T>>(objects_.at(id))->template getValue<...>();
    }
}