$ORACLE_HOME/bin/dbhome

$ORACLE_HOME/bin/dbhome is a shell script that is intended to set the value of the environment variable $ORACLE_HOME to the correct value of the directory where the Oracle Home of a given Oracle SID is located:

ORACLE_HOME=$(dbhome SID)

The script finds the value for the Oracle Home in /etc/oratab.

The script is copied from $ORACLE_HOME/bin to the directory pointed at with the environment variable LBIN (typically /usr/local/bin) by $ORACLE_HOME/install/utl/rootinstall.sh.