This question is about the Barechu, the Call to Prayer in Shacharit. When the shaliach tzibbur says "Barechu" he bows, then straightens up at "Adoshem". Similarly, the congregation in response bows when they say "Baruch" and straightens up at "Adoshem". Then the shaliach tzibbur repeats their response (so he can bless Hashem, too), again bowing when he says "Baruch" and straightening up at "Adoshem". Is all of this correct? I would like especially to confirm that the shaliach tzibbur bows a second time when he repeats the congregation’s response. Thank you for your help.

Shalom, Thank you for your question. The way you described all the bowings for Barachu is perfectly correct. Here are some additional sources for you. The Biur Halacha (in the Mishna Brurah) 113, 3 writes that though there are those who question the source for bowing at Barachu, “in truth this custom has a basis from the verses in Chronicles 1, 29, 20 (“And David said to the entire assembly, "Now bless the Lord your God," and the assembly blessed the Lord, the God of their fathers, and they kneeled and prostrated themselves before the Lord and before the king.) And the customs of Yisrael are [considered as] law”. However, there are those who argue on this custom – and because it is not mentioned in the Gemara, the Vilna Gaon (Gra) holds that there are no bows during Brachu at all. (In fact the Gemara suggests that someone who adds bowing where the Rabbis did not institute it is acting badly). Some communities (as well as some Sephardi communities, following the ruling of the Or Le'Zion 2, 85:13) follow this practice and refrain from bowing, or at least only the Chazan bows. The Aruch HaShulchan (113, 6) suggests that one should not bow fully as one does in the Amidah, but rather a smaller bow. In any event, the widespread practice is to bow – as you wrote. Because the source is based on the verse that indicates one should bow when blessing Hashem, it follows that the Chazan should bow again when repeating the congregation's response, as that too is a blessing of Hashem. Also, the reason he repeats it is to include himself with the congregation – it therefore makes sense that he should include himself not only in the words the congregation said, but also their action (of bowing). You are also correct that we should say Hashem's name when standing straight. This is true of all the bowing we do also in the Amidah. I hope this is of some help to you - Blessings.