The South African's team have the opportunity to claim the trophy as they head into Saturday's fixture with Edinburgh holding the advantage from last weekend.
"Obviously it would be fabulous to win the 1872 cup but our objective is much bigger, Smith said.
"We dream of winning the competitions and we would like to be the best versions of ourselves every week and at the back end of the competition and leading into next season.
"We also want to give the Scottish boys the best preparations going into the Six Nations. There are other objectives for us, obviously winning the cup would be massive for us, we'd really enjoy that but for us we need to focus on our processes if we want to extend our challenge into the next part of the season.
When asked about the emergence of young scrum-half Ben Afshar, Smith added: "Ben is looking good, he has improved a lot and I'm always proud of the boys growing in our environment, getting better, evolving and changing perception, firstly their own perception of themselves as players but also people around the team's perception.
"He has grown in confidence, he's got the support of the older players in the team and they're excited to have him as part of the group. That is what we try and do here, give everyone the chance to be world class and starters.
"We have Jamie Dobie coming back from injury and he'll be around with Sean Kennedy going into the six nations. Ben has definitely lifted his arm up for more responsibility and if he gets the chance I'm sure he'll be ready to take it."