Totally fine. All Shimano indexed-shifting rear deraillers (excluding 8-speed and prior Dura Ace) have the same cable pull ratio, so they're compatible. Which means shifters designed for 8 speeds will work with a 10 speed cassette, as long as you have 10-speed shifters.
If your down-tube shifters are Dura-Ace 8-speed, this wouldn't work since they had a diff cable pull ratio. But Ultegra and everything else is on the same compatibility standard (as are Dura Ace 9- and 10-speed, thankfully).
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"c" is not a unit that measures tire width