Usually, the larger honor is felt by the player -- who gets a "once-in-a-lifetime" thrill -- and not the team that selects dozens of players each year.
But because of Azor's character -- he was the captain of the Naval Academy as a senior this past season -- LaMar felt especially proud of his franchise for making the selection.
"Alex plays the game with passion every day," LaMar said. "We chose him because of his ability and his love for the game."
LaMar got the chance to scout Azor extensively, partly because his son, left-hander Charlie LaMar, was a Navy freshman this past season.
The scout watched Azor, 23, lead Navy with a .322 batting average, making the All-Patriot League first team for the second time in his career.
A lefty batter, Azor hit just one homer, three triples and four doubles, but his .419 on-base percentage was third on the team.
Azor also set the Navy record for highest round selected in the annual baseball Draft.
The 5-foot-11, 190-pounder said the Jays called him in the 10th round and asked him if he would be willing to sign on the cheap. "I would have signed for a hot dog," said Azor, who added that his bonus was a modest $1,000.
Azor reported this week to the Jays' rookie-league team in Dunedin, Fla. But just how long he will play is still in question. He has a five-year commitment to the Navy, and he is set to report to the Marine Corps headquarters in Quantico, Va., in April 2013. (The Marine Corps is a component of the U.S. Department of the Navy.)
"My dream is to play baseball, but I also want to serve my country," Azor said. "I feel like I have a debt to my country."
Azor is grateful that the U.S. took in his parents and grandparents. His father, Michel Azor, is a native of Camaguey, Cuba. His mother, Helida, is a native of Cali, Colombia.
Then again, most would agree that Michel Azor, who served in the Vietnam War, has already paid his so-called "debt." But, like most of the men and women in the military, honor is a big word for Alex Azor.
"Alex was the first unanimous team captain we've ever had," Navy coach Paul Kostacopoulos said. "That honor is voted on by the players, and it's almost impossible to have everyone vote for you, especially on a team full of leaders. But Alex manages to connect with everyone."
Kostacopoulos said he expects Azor to play this summer and then begin his military commitment in April. From there, the coach said Azor can apply to be released from the rest of his commitment in April 2015. Azor would be 26 at the time.
"He would still serve in the reserves for six more years, and do recruiting for the Navy," Kostacopoulos said. "It would be an alternate type of service, and the best-case scenario for his baseball career.
Whatever happens, Kostacopoulos essentially said that you shouldn't bet against Azor.
"Every time Alex has been faced with a challenge," the coach said, "he's come out on top."
But even if Azor never makes the Majors, he has already helped the Jays by simply signing. It's all part of the new regulations Major League Baseball put in play for the 2012 Draft.
As part of those rules, each team is assigned an overall budget depending on how many picks it has and how high up it is in the Draft. Toronto's Draft budget this year is $8.83 million.
If the budget is exceeded by five percent or more, the offending team would have to forfeit a first-round pick in the next Draft. If it is exceeded by 15 percent, the penalty is a pair of first-rounders.
Major League Baseball also assigns a dollar amount for each slot in the first 10 rounds. Teams can go over or under slot for each pick -- as long as they stay within the overall budget.
The Blue Jays used their first seven picks this year to draft high-end players. They then used their next seven selections on college seniors, including Azor, who had little leverage in terms of extracting a large signing bonus.
It's a strategy that has earned praise from more than one media outlet. But to LaMar, it's something far more personal.
"We made a dream come true for a young man who loves baseball," LaMar said.
Walter Villa is a contributor to MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.