INNER JOIN collections in Spring Data MongoDB

How do I implement INNER JOIN in Spring Mongo?

Stupid sample just for example, it's actually incorrect, I just want to show many-to-many relation:

@Document(collection = "people")
public class Person {

    @Id
    private String id;
    private String name;
    private String petId;

    // Getters, setters, constructors and etc.

 }

 @Document(collection = "pets")
 public class Pet {

    @Id
    private String id;
    private String name;
    private PetType petType; // Dog, Cat etc.

    // Getters, setters, constructors and etc.

 }

If I want to find all dogs that belong to John Smith, how should I do it? I need a query like this:

SELECT
     pt.*
FROM
     pets AS pt INNER JOIN people AS pe ON (pt.id = pe.petId)
WHERE
     pt.petType = ${input_petType}
     AND pe.name = ${input_name}

It means that I have two conditions in collection Pet and collection Person:

import org.springframework.beans.factory.annotation.Autowired;
import org.springframework.data.domain.Pageable;
import org.springframework.data.mongodb.core.MongoTemplate;
import org.springframework.data.mongodb.core.aggregation.Aggregation;
import org.springframework.data.mongodb.core.aggregation.LookupOperation;
import org.springframework.data.mongodb.core.query.Criteria;
import org.springframework.data.mongodb.core.query.Query;

import java.util.List;

public interface PetRepository extends MongoRepository<Pet, String>, PetCustomRepository {

}

public interface PetCustomRepository {

     List<Pet> findAllByPetTypeAndPersonName(PetType type, String personName, Pageable pageable);

}

public class PetCustomRepositoryImpl implements PetCustomRepository {

    @Autowired
    private MongoTemplate mongoTemplate;

    @Override
    public List<Pet> findAllByPetTypeAndPersonName(PetType petType, String personName, Pageable pageable) {
        LookupOperation lookup = LookupOperation.newLookup()
                 .from("people")
                 .localField("_id")
                 .foreignField("petId")
                 .as("join_people");
        Aggregation aggregation = Aggregation.newAggregation(
                 Aggregation.match(Criteria.where("petType").is(petType)),
                 lookup,
                 Aggregation.match(Criteria.where("join_people.name").is(personName)),
                 Aggregation.skip(pageable.getPageNumber() * pageable.getPageSize()),
                 Aggregation.limit(pageable.getPageSize()));
        return mongoTemplate.aggregate(aggregation, Pet.class, Pet.class).getMappedResults();
    }

}

The findAllByPetTypeAndPersonName() method returns empty list. What am I doing wrong?