10 μm is fairly unusual. However, you should be able to connect them directly using a SMF patch cable. LC vs SC isn't a problem, but the core difference might be.
The key is your link power budget. While it doesn't hurt to go from 9 μm to 10 μm (with 1500 nm), the opposite direction will suffer some attenuation (from reducing the cross section). I'd expect log(92/102)*10 = -0.9 dB.
If that's within the power budget, everything will be all right. With OS2 grade fiber, the attenuation is .4 dB/km, so the adaption costs you ~2 km reach. If the power budget is exceeded, you'll need a 10 μm patch cable and link that directly into a transceiver, in a switch or in a media converter.