Firstly, whenever possible one should spawn fish in a species tank. Most other fish see the eggs or fry as food and will eat them as fast as they can find them. Even the breeders themselves will eat the eggs.
Secondly, you can remove cory eggs not long after they are pasted to things and then hatch and raise them in a separate space/tank. This requires using Methyl Blue or other fungicide which should be removed from the water once to spot wigglers. You would need a heater, sponge filter or airstone and air pump as well as a heater. A java moss rock or a other plants can be added once the fry are almost free swimming as a source of infucoria. Freshly hatched or frozen bbs or frozen cyclop-eeze are great starter foods. But any good quality fry food should work.
Third, the temperature of the water is really a secondary concern in triggering. Understanding the process involves understanding what goes on in the wild and why one is trying to replicate this in a tank. Corys are seasonal spawners. There is a dry and a rainy season. The key differences are the conductivity and TDS of the water and the availability of food. Flow rates and temperature changes are the secondary cues.
In the dry season the pH is higher, the temperature is higher, the flow is slower, water levels are lower and food is more scarce. The onset of the rainy season is heralded by a big barometric pressure drop- and this is a big trigger as it alerts the fish to the impending change in seasons. This is like the loudspeaker on the platform announcing the train is about to arrive. And then the rains come and things change. There is an inrush of cleaner water much lower in conductivity/TDS, lower in pH and lower in temperature. And with this comes lots of good food. It is the change in seasons that acts as the trigger.
So how does one trigger the fish. Timing a big water change to the arrival of a storm helps a ton, much more so than just cooler water. Good high protein food made plentiful is also important. To simulate the dry season one can minimize water changes and allow the conductivity/TDS levels to rise. You are basically letting your water get dirtier but not letting nitrates become and issue. you can also achive this chemically with a mix of Calcium carbonate (or crushed coral). epsom salts and a small amount of baking soda. Both methods make your tap water the rainy season and your raised levels the dry. The reverse is to use some amount of RO water to create the rainy season and let your tap and fewer water changes or the additives be the dry.
It is normally not necessary to get into the temperature stuff and the flow rates unless the fish prove to be stubborn and then you may not only have to resort to that but a few other tricks as well. One is to take water from a tank in which other corys have just spawned and put it into a tank you are trying to trigger the fish in.
These methods are used for almost any of the seasonal spawners. I use it on my plecos and used to on corys.
But here is the best part. usually the trick is to get the first spawn. Most seasonal spawners will continue to spawn once they have gone the first time and it is why many folks report "they just went by them selves without my doing anything." They will take extended breaks but will also usually restart with little effort needed after most hiatuses.
Hope this all helps some.
