6:21 Ἵνα δὲ εἰδῆτε καὶ ὑμεῖς τὰ κατ’ ἐμέ, τί πράσσω, πάντα γνωρίσει ὑμῖν Τύχικος ὁ ἀγαπητὸς ἀδελφὸς καὶ πιστὸς διάκονος ἐν κυρίῳ,
Ἵνα δὲ εἰδῆτε καὶ ὑμεῖς – The particle δέ marks a new topic. The purpose clause precedes the main clause and thus receives some emphasis. There is probably a chiastic structure in vv. 21-22, with the initial ἵνα…εἰδῆτε picked up by ἵνα γνῶτε in v. 22. The subject of the verb is the emphatic καὶ ὑμᾶς with the ascensive καί.
τὰ κατ’ ἐμέ — This is the direct object, that is further qualified by the following interrogative. In this construction the article nominalizes the prepositional phrase.
τί πράσσω – This is an indirect interrogative clause which functions appositionally with τὰ κατ’ ἐμέ. The interrogative is the direct object of the verb.
πάντα γνωρίσει ὑμῖν – The pronominal adjective πάντα references τὰ κατ’ ἐμέ and is the object of the future active indicative verb. The dative marks the indirect object.
Τύχικος ὁ ἀγαπητὸς ἀδελφὸς καὶ πιστὸς διάκονος ἐν κυρίῳ — Paul takes pains to identify and recommend Tychikos. Two appositional structures define Tychikos as ἀδελφὸς…διάκονος, both of which also are modified by attributive adjectives. The adverbial prepositional phrase indicates the sphere within which this relationship exists.
6:22 ὃν ἔπεμψα πρὸς ὑμᾶς εἰς αὐτὸ τοῦτο, ἵνα γνῶτε τὰ περὶ ἡμῶν καὶ παρακαλέσῃ τὰς καρδίας ὑμῶν.
ὃν ἔπεμψα πρὸς ὑμᾶς – This relative clause offers further information about the role of Tychikos. The adverbial prepositional phrase indicates motion and thus has a spatial sense.
εἰς αὐτὸ τοῦτο — This adverbial prepositional phrase indicates the purpose for which Paul sent him, i.e., “for this very thing” and is an anticipation of the content of the following ἵνα clause.
ἵνα γνῶτε τὰ περὶ ἡμῶν – Compound purpose clause. The object is the nominalized prepositional phrase and the preposition indicates reference. It is very similar to the prior τὰ κατ’ ἐμέ. Here Paul uses the plural form of the pronoun indicating that others are with him during his imprisonment in Rome.
καὶ παρακαλέσῃ τὰς καρδίας ὑμῶν – The verb means “to comfort, encourage.” τὰς καρδίας is the direct object.
6:23 Εἰρήνη τοῖς ἀδελφοῖς καὶ ἀγάπη μετὰ πίστεως ἀπὸ θεοῦ πατρὸς καὶ κυρίου Ἰησοῦ Χριστοῦ.
Εἰρήνη τοῖς ἀδελφοῖς καὶ ἀγάπη – In salutations normally main verbs are implied and not used. They often are in the form of prayers or wishes, i.e., “may peace and love belong to the brothers.” The dative could mark possession or perhaps advantage. The compounded nouns “peace and love” perhaps summarize what Paul has been discussing in his letter. Presumably the source of this peace and love is God.
μετὰ πίστεως — Paul adds a third element in μετὰ πίστεως. There is no article with the noun and so in this context it may describe the gift of faith as in 2:8-10. The preposition would indicate association.
ἀπὸ θεοῦ πατρὸς καὶ κυρίου Ἰησοῦ Χριστοῦ — The preposition probably indicates source. Both God and the “Lord” are noted jointly as the source. πατρὸς…Ἰησοῦ Χριστοῦ are appositional to their respective nouns.
6:24 ἡ χάρις μετὰ πάντων τῶν ἀγαπώντων τὸν κύριον ἡμῶν Ἰησοῦν Χριστὸν ἐν ἀφθαρσίᾳ.
ἡ χάρις – This nominative is the subject of this second prayer or wish, with an implied verb as in the previous verse.
μετὰ πάντων τῶν ἀγαπώντων τὸν κύριον ἡμῶν Ἰησοῦν Χριστὸν – The adverbial prepositional phrase expresses the idea of association. The substantival participle is the head noun of this construction. The present participle tense form may suggest in this context a continuing activity. The direct object is τὸν κύριον, which has the appositional Ἱησοῦν Χριστόν. The pronoun is a genitive of subordination.
ἐν ἀφθαρσίᾳ — This adverbial prepositional phrase probably modifies the participle τῶν ἀγαπώντων and describes the manner of the love being expressed, i.e., one that is incorruptible and therefore immortal.