  Coronal jets are short-lived eruptive features commonly observed in polar
coronal holes and are thought to play a key role in the transfer of mass and
energy into the solar corona. We describe unique contemporaneous observations
of a coronal blowout jet seen by the Extreme Ultraviolet Imager onboard the
Solar Orbiter spacecraft (SO/EUI) and the Atmospheric Imaging Assembly onboard
the Solar Dynamics Observatory (SDO/AIA). The coronal jet erupted from the
south polar coronal hole, and was observed with high spatial and temporal
resolution by both instruments. This enabled identification of the different
stages of a breakout reconnection process producing the observed jet. We find
bulk plasma flow kinematics of ~100-200 km/s across the lifetime of its
observed propagation, with a distinct kink in the jet where it impacted and was
subsequently guided by a nearby polar plume. We also identify a faint faster
feature ahead of the bulk plasma motion propagating with a velocity of ~715
km/s which we attribute to untwisting of newly reconnected field lines during
the eruption. A Differential Emission Measure (DEM) analysis using the SDO/AIA
observations revealed a very weak jet signal, indicating that the erupting
material was likely much cooler than the coronal passbands used to derive the
DEM. This is consistent with the very bright appearance of the jet in the
Lyman-$\alpha$ passband observed by SO/EUI. The DEM was used to estimate the
radiative thermal energy of the source region of the coronal jet, finding a
value of $\sim2\times10^{24}$ ergs, comparable to the energy of a nanoflare.
