We are going to look at the meaning of the 18th and 19th Mantras of the first adhyaya of the Prathama valli of Kathopanishad.

त्रिणाचिकेतस्त्रयमेतद्विदित्वा
य एवं विद्वाँश्चिनुते नाचिकेतम् ।
स मृत्युपाशान् पुरतः प्रणोद्य
शोकातिगो मोदते स्वर्गलोके ॥ १८॥
As the second boon to Nachiketas, Yamaraja taught him the Swargya Agni vidya, the Chayana vidya through which one goes beyond the mortal worlds.
I this mantra, Yamaraja is just reiterating that someone who knows the Trinachiketa, he cuts the ropes of death ahead of him, goes beyond all sorrows and enjoys in Swargaloka.
Basically, he goes beyond the cycle of births and deaths.
There will not be any rebirth for him.
We have seen in detail about this Agni vidya or going beyong the mortal worlds by means of performing yajas in the previous mantras.

Now, the 19th mantra -
एष तेऽग्निर्नचिकेतः स्वर्ग्यो
यमवृणीथा द्वितीयेन वरेण ।
एतमग्निं तवैव प्रवक्ष्यन्ति जनासः
तृतीयं वरं नचिकेतो वृणीष्व ॥ १९॥

Yamaraja is telling Nachiketas - I have told you about the Agni Vidya as the second boon.
As an additional boon, as a fourth boon, I also named this vidya after you, the Trinachiketagni vidya.
Now, you ask for the third boon which I had promised you.

Nachiketas said -
This is the 20th mantra -
येयं प्रेते विचिकित्सा मनुष्ये-
ऽस्तीत्येके नायमस्तीति चैके ।
एतद्विद्यामनुशिष्टस्त्वयाऽहं
वराणामेष वरस्तृतीयः ॥ २०॥

Some say that body and atma are separate. After dying the atma goes to another world to undergo the results of its karma.
Some say that there is no atma different from the body. When the body perishes, atma also perishes.
This dispute and debate have been going on ever since creation.
A mortal being can not answer this question.
As my third boon, please settle this debate for ever and give me a clear answer.

Here, one thing is clear.
Nachiketas went to Yamaloka not after dying.
He had only fainted.
Otherwise, he himself would have known the answer to this.

Now, you know these terms, Astika and Nastika, beleivers and non-believers.

In the mantra,
अस्तीत्येके नायमस्तीति चैके
Some say Asti and some say Nasti.
Asti means - yes, it is there.
What is there?
Atma.
Nasti - No, it is not there, Atma is not there.
This is where these terms have come from.
It could be Ishwara also - Ishwara is there, Ishwara is not there.

So this is the most important question - is there something other than the body?
This is what Yamaraja is going to clarify.
This is the third boon to Nachiketas.