Garudan was the archaryan to Swami Desikan. Swami Desikan prayed to Garudan in Aushada giri in Thiruvendhipuran, who gave the Swami the moola mantharam and the idol of Laxmi Hayagrivar.
It was Garuda Seva time in Kancheepuram and the temple priest received a divine message that He would disappear from their midst for a few minutes to appear before Doddacharya in Sholingar and He did so. To commemorate this unique event, the Lord's idol, during the Garuda Sevai, is hidden from public view with two huge umbrellas for a few minutes and they are removed after camphor is lit and Harathi is performed to the Lord.
Generally in Garuda vahanam, Emperuman would rest both his foot in Garudan’s hands. The Garudan in moolavar, in this temple, is quite unique as Emperuman is seen resting his right leg on the shoulders of Garudan and the right hand of the Garudan is in chin mudra. As per the current Doddacharya of the 23rd generation, H.H. K.K.C. Singarachari, the reason for this mudra is to realize the lord one has to be guided by an acharyan and in the case of Doddacharya it was Garudan who was the acharyan for Doddarcharya.