This is the Emir Sultan Mosque, close to the Meryam Kilesi.

This small mosque was built in 1397, together with the türbe (tomb) of Hoca Izzeddin (= Emir Sultan) next to it.
The date corresponds to the very end of the Kadi Burhanettin Beylik rule (1381-1398). This explains the Seljuk-type pyramidal roof of the türbe, since Kadi Burhanettin placed himself in the former Rûm Seljuk line and cultural traditions (his grandmother was the granddaughter of sultan Keykavus II).
Both buildings are small: 4 x 4 m for the tomb, 6 x 10 m for the prayer hall of the mosque. The structure in the right upper corner is called a ‘köşk minare’ (kiosk minaret) or also ‘minber minare’ (pulpit minaret). Mosque and tomb were repaired in 1980.

Kadı Burhaneddin was vizier and atabeg to the Eretna rulers of Sivas - Kayseri. In 783 AH (1381/82) he took over Eretnid lands and claimed the title of sultan for himself. He is mostly referred to by the title “kadi“ (= Islamic judge), his first position under the Eretnids. He met his end in 1398 fighting the Akkoyunlu and was succeeded by his son Zayn al-‘Abidin, who ruled for a short time (1398 – 1399). His türbe survives in Sivas.

Correspondent: J.M.Criel, Antwerpen.
Sources: ‘Tarihi Kayseri Cami ve Mescidleri’ – Doç. Ilhan Özkeçeci (1997) & Wikipedia .