I have tried unsuccessfully to establish if this is the head of a bull or cow and why it is here in St Mary’s Street just up from the Cowgate.  The Cowgate used to be a road lying outside the city walls providing a route for cattle from the south and east of Edinburgh to the cattle market near the Grassmarket  below the castle and St Giles Cathedral.  Perhaps the head was above a butcher shop or a dairy.
Without refrigeration and modern ways of treating milk there was a need for cows to be kept near the city for fresh milk.   I read that were 2,085 cows in the city in 1865 and 206 dairies with 3,150 cows in 1898.  The city at that time was very much smaller than it is now since it has incorporated surrounding villages.  This article was written in 1905.
"The condition and surroundings of many of these byres has astonished me. Surely it is a disgrace to Sanitary Science that cowsheds should still exist in the centre of towns. There are many of these still in Edinburgh, and these are often situated in densely populated localities. At present there are 115 such city cowsheds, licensed to contain 3,147 cows. On 1st December 1903 the number of cows in the city amounted to 3,072. In many cases byres containing from 20 to 40 cows are situated behind lofty tenements, and approached by narrow lanes and alleys. In fact they are hidden away in most cases, so that they have to be sought for. The number of cows kept by one dairyman in one locality may reach as high as 47. Of necessity the animals are never allowed out from such city sheds. From the time they are bought as milk-giving cows until they are sold again, they are kept closely confined. Not only is this the case in towns, but the same holds in suburban districts. I know of byres on the outskirts of the city containing large numbers of cows, not one of which is ever outside the sheds. The requirements of the city of Edinburgh in respect of milk are by no means supplied by the city byres, and altogether in Edinburgh, Leith, and in the surrounding districts, there are about 25,000 cows which supply milk for the wants of the populance."     -- Dr W G Aitchison Robertson, Considerations Relating to the Improvement of Milk Supply, Transactions of the Medico-chirurgical Society of Edinburgh, 1905