I'm recently reading on the arguments that for the category of left $G$-sets with the canonical topology. The sheaves are all representable.
To understand the proof, I needed the fact that for any sheaf, if we evaluate it on the empty set, then it must yield the terminal object.
The previous page How to prove that for a sheaf functor F, F(empty set) = terminal object? was really helpful but I couldn't justify some points by myself.
I think I can accept the existence of the product over empty set.
But I don't know how to justify the existence of two morphisms for the equalizer condition of the definition of the sheaf.
Of course, to understand it, I guess we can just use the the vacuousness of the existence of pair of indices of the empty index set.
But I think we must be able to describe the sheaf condition more categorically.
For example, we may bypass the existence of a pair of elements of the index set since that is vacuously true. But the two maps in the sheaf conditions are induced by the projections of the fiber product.
Does vacuous pair of indices also gives us their fiber product?
What I want is a nice and practical and axiomatic approach(new definition that is just consistent) that can 'just' (practically or axiomatically) embrace the empty index sets. (the empty covering)
So to summarize I don't know how to justify the existence of two morphisms.
Thank you for reading this question!