For every whole number version (i.e. LR5.x, LR6.x), the catalog database structure changes to incorporate new features. For this reason you can not open an LR6 catalog file in LR5 You can convert a LR5 catalog file to the LR6 structure. And this is what happens when you upgrade to a new version. It is a "one-way" street. You can not convert a LR6 catalog file to the LR5 structure.
Do you own the license for LR on both machines? If you do you can install LRCC on the office computer as two installs are within the terms of your subscription license. If you have LRCC installed at home, you can install a LR5.7.1 license along side it. But doing this means that you don't get any of the added functionality that LRCC brings.