The Mezquita is the clearest European example of a sacred building that changed religion by conquest and has stayed changed, and of how unresolved that can remain eight centuries later.
The building
Begun in 785-786 under Abd al-Rahman I, reportedly on the site of the Visigothic church of St Vincent, though the traditional account of a shared building bought out in stages rests on thin archaeology. Four successive expansions carried it to its present extent by the end of the tenth century. The hypostyle hall of double horseshoe arches in red brick and white stone, and the mihrab with its gold mosaics sent by a Byzantine emperor, make it the outstanding surviving monument of al-Andalus.
1236: Ferdinand III of Castile took Cordoba and the mosque was consecrated as a cathedral within months.
1523: a Renaissance cathedral nave was driven up through the middle of the hall. Charles V is traditionally said to have regretted approving it, the remark attributed to him, that they had built what could be built anywhere and destroyed what was unique in the world, is repeated everywhere and documented nowhere contemporary. Treat it as a good story.
The ownership question
In 2006 the Catholic Church registered the building in its own name in the Spanish property register, for a nominal sum, using a provision of mortgage law that permitted the Church to register property without producing a title deed, a mechanism originally designed for parish churches and since amended.
Cordoban civic groups, academics and regional politicians objected that a monument of that significance, maintained in part with public money and visited by close to two million people a year, is public patrimony and cannot be converted to private ownership by a registry filing. Petitions gathered hundreds of thousands of signatures. Litigation and political attempts to reverse the registration have not dislodged it.
The prayer question
Muslim prayer inside the building is prohibited. The prohibition is the Church's, and it has been maintained through repeated requests from Spanish Muslim organisations and from bodies abroad, and through at least one confrontation in 2010 in which a group of visitors prayed inside and security staff intervened, with injuries and arrests.
The Church's position is that the building is a consecrated cathedral, that shared worship in one space is not workable, and that it has no obligation to reintroduce a use ended in 1236. Those asking cite the building's origin and the coexistence Cordoba is famous for. Spanish Muslim organisations have generally framed the request as symbolic; Church authorities have generally treated it as a claim on the building.
The name
Church signage and publications have moved toward calling the building simply the Cathedral of Cordoba, or the Cathedral of Our Lady of the Assumption, and away from the compound Mosque-Cathedral. Campaigners read the shift as an attempt to write the mosque out of the building's description; the diocese replies that it is accurate about what the building now is.
Nobody disputes any fact in the paragraphs above. The disagreement is entirely about what follows from them.