Gin and Shuichi Akai

Gin and Akai Shuuichi are rivals and hate one another.

Background

A five years prior to the current timeline, FBI agent Akai Shuuichi infiltrated the Black Organization using the alias Dai Moroboshi as a mole and dated Akemi Miyano, one of their low-ranking members, in order to meet Shiho Miyano and use her influence to get in. Although at first he was simply using Akemi as a tool, he really fell in love her and she with him in return. Akai distinguished himself as an agent and two years prior to the current timeline received the codename Rye and was called to work under executive agent Gin. The FBI laid a trap hoping to capture Gin at a warehouse where the meeting was supposed to take place, but the Organization discovered Akai's true allegiance and Gin never appeared.

Current Timeline

Although Akai cut off contact with Akemi afterwards, the Org remained afraid that Akemi could be used against them by Akai but unable to eliminate her because the Org needed the continued loyalty of her sister Miyano Shiho because of her work on the APTX 4869 project. Just before the current timeline, the Org came up with a plan they thought would enable them to remove Akemi Miyano from the Organization without jeopardizing Shiho's loyalty. Although they had no intention of actually doing so, the Org. promised Akemi that she and her sister would be freed if she could successfully pull off a 1 billion yen bank robbery. The Org expected Akemi to fail which would provide the excuse to kill her, but Akemi succeeded and then was killed for no reason by Gin.

Later, Akai returned to Japan with the FBI with the goal of capturing Vermouth. At this time Akai stated cryptically to James Black that his loved one (koibito) had dumped him and Akai had returned with the intention of making the loved one regret ever leaving him with "tears of blood". Later, Akai, set up 700 yards away, snipes one of Conan's bugs out from between Gin's fingertips. Gin, perhaps anticipating who would be responsible, grabs Korns rifle and uses it to locate Akai Shuuichi. At this time Akai fires a bullet that passes through Gin's scope and grazes his left cheek causing it to bleeds like he is crying. At this time Akai says "we finally meet, my lovely, lovely koibito-san". At this time, Aoyama Gosho uses a japanese literary device to give the word "koibito" a double meaning. Japanese in Detective Conan includes the pronunciation, called "furigana", in hiragana next to the kanji. Normally the furigana matches the word in the kanji; however, in this case Gosho used the kanji for "old enemy" (宿敵), but the furigana for "koibito" (こいびと), thus giving what Akai said a double meaning. The true meaning of what Akai meant is "old enemy", but what he said out loud was the furigana "koibito". In effect Akai had nicknamed Gin "loved one".

Akai then fired twice into Gin's bullet proof vest to encourage him to leave quietly, purposefully avoiding killing Gin although he was capable of finishing him. Akai's stated reason for letting Gin go was to avoid a gunfight erupt in the middle of Beika city. Despite Gin having enough internal damage to bleed from his mouth, he still manages to smirk as he leaves, realizing they had been set up.

Analysis

Gin seems to have a lesser opinion of the damage Akai could cause to Org compared to the boss and Vermouth's impressions. Vermouth and the boss fear that Akai could become a silver bullet while Gin doesn't believe a single person who could take down the Organization exists.

Before Akai Shuuichi received his codename "Rye" and was called up to work under Gin, it has isn't clear if Akai had previously worked with Gin directly, despite their current personal antipathy for one another. If the Organization made preparations like sending in a scout ahead of time, it seems more like a first time meeting rather than a re-meeting up. Kir didn't have to go through checks of that level before her mission.
There are a couple ways that Gin and Akai cold have developed a rivalry with Akai despite not apparently working together while in the Org. It is known Akai survived multiple attempts on his life after he left, so if Gin was responsible for some of these attempts, perhaps the rivalry developed when they tried to take each other out during the two years after the FBI mission failed. Another possibility is that Gin and Akai met each other via Shiho Miyano as Akai was getting into the Org. Akai used Akemi to get close to Shiho. If Gin was in a relationship with Shiho at the time, then Gin could have met Akai Shuuichi as he was dating Akemi and meeting with Shiho.

In the Red Shirt's case while Gin was staking out the Beika department store waiting for scar Akai to appear, he replies to Kir's remark that "Why don't you [Gin]... just enjoy the fact your nemesis is gone?" with "Yeah, I'm so happy... If he is still alive, I get to kill him again." This shows that Gin considers Akai's death as "his kill" even though it was Kir who pulled the trigger at Raiha pass, and if scar Akai is killed, it will be Chianti or Korn who finishes the job. He is apparently satisfied with the orchestrator role and doesn't feel the need to personally kill Akai himself.