They visited the Temple of Fortuna, the goddess of fortune, known as Tyche to the Greeks.
Hadrian and Antinous made sacrifice to her for the benefit of the tour and for the Empire as a whole. Fortuna, also known as Bona Dea, a manifestation of the Great Mother as a girl, in her aspect as the bestower of goodness, wealth and prosperity.
Her consort was Agatho Daimon, the "Blessed Daimon." In Greek, "daimon" means "spirit-being" regardless whether that being is beneficial or detrimental to humans. It was only later that the Christians literally "demonized" the word daimon, so that the modern word "demon" only ever refers to an evil spirit.
Agatho Daimon is a decidedly beneficent spirit, rendered often as a serpent.
Antinous was later compared to Agatho Daimon, and is often portrayed with the Serpent in his hand, as in the famous Berlin Antinous-Agathodaimon statue pictured here.
The arrival of the court in the city of Corinth signaled the beginning of the association between Antinous-Agatho Daimon and his female counter-identity as Fortuna.
We consecrate this day to our own fortune and to the forces of chance that play through our lives, as Hadrian and Antinous submitted their hope to the goddess at Corinth.