The affix vuñ comes in the sense of 'nature or action thereof' after a gotra (Patronymic\/Family-name) and after the Name of a Vedic School when one boasts thereby or manifests his contempt thereby or when it means one who has attained that (or has come for an inspection or inquiring thereof).,
The word श्लाघा means boasting; अत्याकार means 'contempt', तद् अवेतः means तद्प्राप्तः 'who has gained that', or तज्ञ्ज्ञः 'who inquires that'. The word तत् refers to गोत्र and चरण ॥ The word तदवेतः means \who has obtained that or who has known that\.
To take first श्लाघा as गार्गिकया श्लाघते or काठिकया श्लाघते \boasts of belonging to the family of Garga, or the school of Katha\. To take अत्याकारः, as, गार्गिकयात्याकुरुते, काठिकयात्यकुरुते \he manifests contempt upon another, because of his belonging to the clan of Garga or School of Katha\. To take तदवेता as, गार्गिकामवेतः, काठिकामवेतः = प्राप्तः or अवगतवान् ॥
Why do we say \when it means boasting &c\ Observe गर्ग्यत्वम्, कठत्वम् ॥ The य of गार्ग्य is elided in the above examples by 6.4.148 & 6.4.151.,