Exegesis:

kai katēlthen ‘and he went down.’ kai connects the following narrative with the preceding and marks it as a continuation.

katerchomai (also 9.37) ‘to go down’ or ‘to go a lower place.’ Here the difference in height is approximately 1900 feet, that in distance, about 20 miles.

Kapharnaoum polin tēs Galilaias ‘Capernaum, a town of Galilee.’ polin tēs Galilaias is added as an identification. Capernaum was mentioned already in v. 23 but this first mentioning neither gave opportunity for nor required a geographical explanation.

kai ēn didaskōn autous en tois sabbasin ‘and he taught them on the sabbath.’ Two interpretations are possible which depend ultimately on the understanding of vv. 31-37 as a whole: (1) the clause refers to Jesus’ consecutive teaching on several sabbath days; this also applies to v. 32 which describes the reaction of the people to this teaching. Then vv. 33-37 refer to what happens on presumably the last of these sabbath days; (2) the reference is to a specific sabbath and this clause and the clauses in the imperfect tense in vv. 32f, as contrasted with the aorist tense of anekraxen ‘he shouted’ and subsequent verbs in vv. 32-36, describe as it were the setting in which the sudden outburst of v. 34 happens. The fact that v. 33 begins with kai and does not contain a reference or hint to a specific sabbath to be distinguished from sabbaths referred to previously, points to the second interpretation as being the more natural. This is also consistent with Mk. 1.21-23.

autous refers to the inhabitants of Capernaum who were present in the synagogue, though they have not been referred to previously and the synagogue itself is not mentioned until v. 33, though implied in “was teaching”.

Translation:

He went down. For movement to another level, cf. on “went up” in 2.4.

Capernaum, a city of Galilee, or, ‘(the city) C. (which is) in (the region) G.,’ cf. on 1.26.

And he was teaching them on the sabbath, or, to bring out interpretation (2), ‘once he was teaching them (or, the people (there)) on the sabbath,’ ‘on a certain sabbath he was … (etc.).’ That this happened in the synagogue may have to be stated explicitly.

Quoted with permission from Reiling, J. and Swellengrebel, J.L. A Handbook on the Gospel of Luke. (UBS Handbook Series). New York: UBS, 1971. For this and other handbooks for translators see here . Make sure to also consult the Handbook on the Gospel of Mark for parallel or similar verses.