I have a query around "generalized element" or "probe". If we are discussing a small Set as a category, I could understand that there are could be multiple morphism (function) from set A to X to pick each element in X. But the Category theory is more abstract where we don't know what is inside the category object.

So if I don't know what is inside X, how can I define a morphism to pick each element using probe object? Isn't it a catch-22 position? I can't look into the object X but at the same time I can define a "n" morphism to pick n element. It's confusing me