The term tadīya-śeṣatva is composed of tadīya (belonging to Him, i.e., belonging to Bhagavān's devotees) and śeṣatva (the state of being a śeṣa — a dependent, servant, remainder). Together they denote the obligation of a Śrī Vaiṣṇava to serve and honour all who belong to Bhagavān, not merely Bhagavān Himself.
In Srivachana Bhūṣaṇam, Piḷḷai Lokācārya gives this concept its most complete articulation. He teaches that the śiṣya who has understood true prapatti does not limit kaiṅkarya to direct service at the divine feet; he recognises that Bhagavān's affection for His devotees is even greater than His love for Himself (svasmin api bhagavat-parikare prītiḥ adhikā). To slight a bhāgavata is therefore a far graver offence than to slight the Lord directly. Conversely, to serve and cherish a bhāgavata is the highest expression of devotion.
This principle grounds the Śrī Vaiṣṇava ethic of community. It is the reason the ācārya-śiṣya bond, the care for fellow satsaṅgis, and reverence toward elder devotees are treated as non-negotiable. Tadīya-śeṣatva is what prevents prapatti from becoming a merely private spiritual transaction and situates the prapanna within a living web of mutual service and honour.