Today we will look at the form सौमित्रिः mNs from श्रीमद्-वाल्मीकि-रामायणम् 3.16.3.

स कदाचित् प्रभातायां शर्वर्यां रघुनन्दनः । प्रययावभिषेकार्थं रम्यां गोदावरीं नदीम् ।। ३-१६-२ ।।
प्रह्वः कलशहस्तस्तु सीतया सह वीर्यवान् । पृष्ठतोऽनुव्रजन्भ्राता सौमित्रिरिदमब्रवीत् ।। ३-१६-३ ।।
अयं स कालः सम्प्राप्तः प्रियो यस्ते प्रियंवद । अलङ्कृत इवाभाति येन संवत्सरः शुभः ।। ३-१६-४ ।।
नीहारपरुषो लोकः पृथिवी सस्यमालिनी । जलान्यनुपभोग्यानि सुभगो हव्यवाहनः ।। ३-१६-५ ।।

Gita Press translation – Once that delight of the Raghus, at break of day, went to the beautiful Godāvarī river for the sake of a bath (2). Following at the heels of Śrī Rāma with a vessel for water in his hand together with Sītā, his brave brother, the son of Sumitrā, spoke as follows :- (3) “Now has arrived that season which is dear to you, O polite brother, with which the blessed year appears as though ornamented (4). The people feel dry with cold, the earth is rich with crops; the waters are unenjoyable, the fire is agreeable.” (5)

The above verses have previously appeared in the following post – प्रियंवद-mvs

सुमित्राया अपत्यम् (पुमान्) = सौमित्रिः (लक्ष्मणः) – a (male) descendant of Sumitrā. It refers to Lakṣmaṇa (son of Sumitrā)
In the verses the विवक्षा is प्रथमा-एकवचनम्।

(1) सुमित्रा ङस् + इञ् । By 4-1-96 बाह्वादिभ्यश्च – To denote the sense of अपत्यम् (descendant) the तद्धित: affix ‘इञ्’ may be applied optionally following a syntactically related पदम् in which the सन्धिः operations have been performed, provided the पदम् ends in a sixth case affix and has ‘बाहु’ etc (listed in the बाह्वादि-गण:) as its base. As per the सूत्रम् 4-1-92 तस्यापत्यम् – Following a syntactically related पदम् in which the सन्धिः operations have been performed and which ends in a sixth case affix the तद्धिता: affixes already prescribed (by the prior rules 4-1-83 प्राग्दीव्यतोऽण् etc), as well as those that are going to be prescribed (by the following rules 4-1-95 अत इञ् etc), may be optionally applied to denote the sense of अपत्यम् (descendant.)

Note: First the सूत्रम् 4-1-83 प्राग्दीव्यतोऽण् prescribes the default affix ‘अण्’ which is over-ruled by the affix ‘ढक्’ (prescribed by the सूत्रम् 4-1-120 स्त्रीभ्यो ढक्) which in turn is over-ruled by the affix ‘इञ्’ (prescribed by the सूत्रम् 4-1-96 बाह्वादिभ्यश्च) because ‘सुमित्रा’ is specifically listed in the बाह्वादि-गण:।
Note: Since ‘सुमित्रा’ does not end in the letter ‘अ’ the सूत्रम् 4-1-95 अत इञ् cannot apply here.

Note: आकृतिगणोऽयम्। The बाह्वादि-गण: is a आकृतिगणः – which is a class or group of words in which some words are actually mentioned and there is room left to include others which are found undergoing the same operations.

(2) सुमित्रा ङस् + इ । अनुबन्ध-लोपः by 1-3-3 हलन्त्यम्, 1-3-9 तस्य लोपः।

Note: ‘सुमित्रा ङस् + इ’ gets the designation प्रातिपदिकम्‌ by 1-2-46 कृत्तद्धितसमासाश्च। This allows 2-4-71 सुपो धातुप्रातिपदिकयोः to apply in the next step.

(3) सुमित्रा + इ । By 2-4-71 सुपो धातुप्रातिपदिकयोः – A सुँप् affix takes a लुक् elision when it is a part of a धातुः or a प्रातिपदिकम्।

Note: The affix ‘इञ्’ is a ञित् (has the letter ‘ञ्’ as इत्)। This allows 7-2-117 तद्धितेष्वचामादेः to apply in the next step.

(4) सौमित्रा + इ । By 7-2-117 तद्धितेष्वचामादेः – The first vowel among the vowels of a अङ्गम् which is followed by a तद्धित: affix (ref. 4-1-76 तद्धिताः) which is either a ञित् (has the letter ‘ञ्’ as a इत्) or a णित् (has the letter ‘ण्’ as a इत्) takes the वृद्धि: (ref. 1-1-1 वृद्धिरादैच्) substitute.

Note: The अङ्गम् ‘सौमित्रा’ gets the भ-सञ्ज्ञा by 1-4-18 यचि भम् which allows 6-4-148 यस्येति च to apply in the next step.

(5) सौमित्र् + इ । By 6-4-148 यस्येति च – When the letter ‘ई’ or a ‘तद्धित’ affix follows, the (ending) इ-वर्ण: (letter ‘इ’ or ‘ई’) or the (ending) अवर्ण: (letter ‘अ’ or ‘आ’) of the अङ्गम् with the भ-सञ्ज्ञा takes elision (लोपः)।

= सौमित्रि । The प्रातिपदिकम् ‘सौमित्रि’ declines like हरि-शब्दः।

(6) सौमित्रि + सुँ । By 4-1-2 स्वौजसमौट्छष्टाभ्याम्भिस्ङेभ्याम्भ्यस्ङसिभ्याम्भ्यस्ङसोसाम्ङ्योस्सुप्‌।

(7) सौमित्रि + स् । अनुबन्ध-लोपः by 1-3-2 उपदेशेऽजनुनासिक इत् and 1-3-9 तस्य लोपः।

(8) सौमित्रिः । रुँत्व-विसर्गौ by 8-2-66 ससजुषो रुः and 8-3-15 खरवसानयोर्विसर्जनीयः।

Similarly, we can derive the following –
१) कृष्णस्यापत्यम् (पुमान्) = कार्ष्णिः – a (male) descendant of Kṛṣṇa (of the Vṛṣṇi dynasty)
२) प्रद्युम्नस्यापत्यम् (पुमान्) = प्राद्युम्निः – a (male) descendant of Pradyumna (of the Vṛṣṇi dynasty)
३) सत्यकस्यापत्यम् (पुमान्) = सात्यकिः – a (male) descendant of Satyaka (of the Vṛṣṇi dynasty)
४) युधिष्ठिरस्यापत्यम् (पुमान्) = यौधिष्ठिरिः – a (male) descendant of Yudhiṣṭhira (of the Kuru dynasty)
५) अर्जुनस्यापत्यम् (पुमान्) = आर्जुनिः – a (male) descendant of Arjuna (of the Kuru dynasty)

Note: In the above examples, first the सूत्रम् 4-1-83 प्राग्दीव्यतोऽण् prescribes the default affix ‘अण्’ which is over-ruled by the affix ‘इञ्’ (prescribed by the सूत्रम् 4-1-95 अत इञ्) which in turn is over-ruled by the affix ‘अण्’ (prescribed by the सूत्रम् 4-1-114 ऋष्यन्धकवृष्णिकुरुभ्यश्च) and finally the सूत्रम् 4-1-96 बाह्वादिभ्यश्च brings back the affix ‘इञ्’ since ‘कृष्ण’, ‘प्रद्युम्न’, ‘सत्यक’, ‘युधिष्ठिर’ and ‘अर्जुन’ are specifically listed in the बाह्वादि-गण:।