1) Based on the pics, It appears that the sub-panel has a 100 amp breaker, if so that is good as this is a requirement for a detached bld'g. Detached bldg's require a main breaker sub and subs installed in an attached bld'g are OK to use a main lug sub.
2) Terminate the 2 hot feeds at the main panel via a 100 amp double pole breaker. Identify and tape each hot feed conductor with black and red tape respectively. Tape the neutral feed with white tape at each end. The main panel needs an Equipment Ground Bar for the Equipment Grounding Conductor going out to the sub.
3) At the detached bld'g, you need an 8 foot ground rod using 8 AWG Copper connected to the sub equipment ground bar. The sub needs an equipment ground bar and must be isolated from the neutral in the sub. If the sub has a green bonding screw or copper bonding jumper, this needs to be removed since the neutral must float in a sub and not be bonded to the sub's metal enclosure. Connect the 2 hot feeders from the main panel to the hot bus bar lugs on the sub. Connect the neutral feeder at the sub to the neutral bus bar. Connect the Equipment Ground Conductor from the main, to the Equipment Ground Bar in the sub. Connect any bare copper Romex grounds to the Equipment Ground Bar in the sub. Do NOT mix grounds and neutrals in the sub's neutral bus bar, they must be separated.
4) The voltage drop for the run using 2 AWG copper is only 5.82 volts which is OK. 4 AWG for the Ground is OK but you only needed 8 AWG. Not sure if you will run Underground Feeder or conduit to the detached bld'g. If UF cable - OK. If conduit, suggest using gray electrical PVC. If conduit, the conductors must be moisture resistant using type THWN Copper. I assume you are using copper for your conductors, if using aluminum, let me know. If using conduit, do not use type THHN as these are not moisture proof.
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