The 1-770858-0: This is a 10-position, 2-row vertical header from TE's Mini-Universal MATE-N-LOK series, pitched at 0.163" (4.14mm) on both the mating and row spacing. It's a board-to-cable/wire connector — the shrouded 4-wall housing and locking ramp keep the mating receptacle aligned and latched, which matters when the cable gets tugged or the assembly sees vibration. Rated 600VAC, with a temperature range of -20°C to 85°C and a UL94 V-0 nylon housing.
Pitch, footprint, and what mates with it
The 0.163" (4.14mm) pitch matches the standard Mini-Universal MATE-N-LOK receptacle family. You'll need the corresponding crimp-socket housing and terminals — the header accepts male pins, so the cable side carries the female contacts. The solder tail length is 0.175" (4.45mm), fitting standard 1.6mm PCBs.