The suspect shot and killed Copeland, 58, in what the police department called an "ambush-style situation".
A police officer was shot and killed Monday while serving a warrant.
San Marcos city manager Bert Lumbreras said at the news conference that Copeland's colleagues were emotional after the shooting.
Police are not identifying the suspect until he can be taken before a judge and formally charged, Stapp said.
On Monday afternoon, a man fatally shot a San Marcos police officer.
The warrant names Stewart Thomas Mettz, 51, who remains in an Austin hospital for treatment of a gunshot wound he suffered during a short standoff with the San Marcos/Hays County SWAT team.
According to the release, the officer was serving the warrant with other officers and was wearing a protective vest. He was transported to the Central Texas Medical Center immediately after where he later died.
Regarding police officers killed in the line of duty, ambush attacks account for the most, according to the Federal Bureau of Investigation. He was shot at some point during the standoff, although police said it wasn't clear whether he shot himself or was struck by one of the officers' bullets.
"Ken's a hero. Today was his day off". He leaves behind a wife and four children. "His picture is hanging in different parts of this police department because everyone here loved him". "They are a part of our family as well and they did a great job".
"Along with everyone learning of a San Marcos police officer's murder today, I am deeply troubled and saddened".
Flags will remain at half-staff through sunset on the day of the officer's funeral, Abbott's order said.
Texas Gov. Greg Abbott tweeted the news, and said: "We pray for the family of the San Marcos police officer killed in the line of duty". The men and women in law enforcement put their lives on the line every day to protect and to serve our communities, and we will never forget their sacrifices.