कोः कत् तत्पुरुषेऽचि

Operational rule vidhi

कत् is substituted for कु in a तत्-पुरुष समास when a word beginning with a vowel follows as the second member.

Word by word (padaccheda)

Target कोः ṣaṣṭhī · ekavacanaSubstitute / term कत् prathamā · ekavacanaRight context saptamī · ekavacanaRight context saptamī · ekavacana

Target Ṣaṣṭhī (6th case): names what gets replaced, "in place of X" (1.1.49 षष्ठी स्थानेयोगा).
Substitute / term Prathamā (1st case): in operational rules, the substitute or the thing introduced; in definitions, the term or its referent.
Right context Saptamī (7th case): "when X follows", the operation applies to what precedes (1.1.66 तस्मिन्निति निर्दिष्टे पूर्वस्य). Also a condition or domain ("in the sense of").

Dotted words are technical terms, dashed ones pratyāhāras (names for classes of sounds or affixes). Hover or tap one to see its card.

Full reading

कोः कत्

Pāṇini leaves out words that carry over from earlier sūtras. Expand to see the rule as a reader actually understands it.

As a rewrite rule (sketch from case endings)

कोः → कत् /

Replace koḥ with kat when tatpuruṣe aci follows. Inherited words (anuvṛtti) may add further conditions.

Commentary

कत् is substituted for कु in a तत्-पुरुष समास when a word beginning with a vowel follows as the second member.
As कदजः, कदश्वः, कदुष्ट्रः, कदन्नम् ॥ Why do we say in a Tatpurusha? Observe कूष्ट्रो राजा ॥ Why do we say "when the second member begins with a vowel"? Observe कुब्राह्मणः, कुपुरुषः ॥

Vart:- कत् is substituted before त्रय, as कत्रयः = कुत्सितास्त्रयः ॥

Śrīśa Chandra Vasu, The Ashtadhyayi of Panini (1897), via ashtadhyayi.com.