Correct practical usage of openapi discriminator with simple request/response

I have an endpoint that takes an id. That id fetches something: either a Dog or a Cat; and returns it (as Pet).. It looks like this:

paths:
  /getPet/{id}:
    get:
      operationId: getPet
      parameters:
        - name: id
          in: query
          description: id of pet
      responses:
        200:
          description: happy
          content:
            application/json:
              schema:
                $ref: '#/components/schemas/Pet'
components:
  schemas:
    Pet:
      type: object
      required:
        - petType
      properties:
        petType:
          type: string
      discriminator:
        propertyName: petType
        mapping:
          kitty: Cat
          dog: Dog
    Dog:
      allOf: ...
    Cat:
      allOf: ...

The flow looks something like this: FE requests with id -> BE fetches pet with id -> Returns the retrieved pet back via controller.

The only way I've found to create a polymorphic object is with discriminator, however it requires 2 things.. 1: A discriminator field and, 2: optionally, a mapping field.

This is how I am implementing it in my controller.. It works, but it feels very wrong what I am doing. I am not really making use of the discriminator in any way, nor the mapping I created, nor do I see a way to do so..

override fun givePets(petType: String?): ResponseEntity<Pet> {
        return ResponseEntity.ok(
        when (petType) {
            "Dog" -> Dog("Dog", true, "Harry", 15)
            "Cat" -> Cat("Cat", 27, "Smurfie", 7)
            "Lizard" -> Lizard("Frog", false, "Smithers", 145)
            else -> TODO()
        })
    }

Could someone please advise on the proper way of doing this? I find I am not understanding the OAS spec when I try and put it into practise, despite re-reading it multiple times.