Immediately, the comparisons began with Ware and Lawrence.
"I think it's unfair to compare players like that," Cowboys coach Jason Garrett said. "It's interesting they have the same first name but that's not what we're in the business of doing."
Ware was a player Lawrence often studied. He said he tried to mirror his game after Ware's, stealing some of his pass rush moves. In two years at Boise State, Lawrence had 20 sacks. He had 33.5 tackles for a loss. He recorded 120 tackles, caused seven fumbles and recovered two.
"I'm going to do all I can to become the best and fill those shoes," Lawrence said.
The Cowboys' front seven lacked juice with the losses of Ware and Jason Hatcher. They signed Henry Melton, but the defensive tackle is coming off a torn anterior cruciate ligament. They re-signed Anthony Spencer, but he is coming back from microfracture surgery. They signed Jeremy Mincey and Terrell McClain, but they are viewed more as complementary players.
On the Cowboys' draft board, Lawrence had what Jones called a "first-round pressure player," grade. They saw him as the third-best right defensive end in the draft, behind Jadeveon Clowney, who was the No. 1 overall pick by the Houston Texans, and Anthony Barr, who went No. 9 overall to the Minnesota Vikings.
The Cowboys generated only 34 sacks in 2013. Ware was held to a career-low six, missed three games with a quadriceps injury and battled other injuries during the season. Spencer played in only one game. Hatcher had 11 sacks and was added as an injury replacement to the Pro Bowl.
Quarterbacks were able to pick apart the secondary in part because the Cowboys could generate little pressure. Four quarterbacks threw for more than 400 yards against the Cowboys. Five quarterbacks had four touchdown passes against the Cowboys.
A secondary needs the pass rush.
"The indispensable part, the part we didn't have last year because Ware was injured most of the year, we hope that we can get from this is immediately our second- and third-down pressure player," Cowboys owner and general manager Jerry Jones said. "We felt that deserved a premium in this draft."
Tyrone Crawford was Ware's teammate for two years with the Cowboys. He was part of Lawrence's recruiting pitch to Boise State three years ago.
"I know DeMarcus Ware is definitely a hard player to come in and try to fill his shoes," Crawford said. "He's the best pass rusher to play the game and the best pass rusher I ever played. But I feel like Demarcus has potential to be the best pass rusher in the league. He's lengthy. He's got good moves. He reacts quickly. He's got good speed. I'm just really excited to see what he can do."
Sounds a lot like Ware.
"I'm my own Demarcus," Lawrence said. "I don't like this trying to be nobody else. I'm going to be me. I'm going to do it well."