I revisited this popular tutorial about category theory after some time and I realised that right at the beginning there is a statement about what a category is:
In a category, if there is an arrow going from A to B and an arrow going from B to C then there must also be a direct arrow from A to C that is their composition.
Is that true in a standard category? All this time I was happy with my composition function
$$\circ : \operatorname{Hom}_\mathcal{C}(A,B)\times\operatorname{Hom}_\mathcal{C}(B,C)\to\operatorname{Hom}_\mathcal{C}(A,C)$$
To my knowledge this function never complained when the codomain was the empty set. Was I operating in something that is not a category? Or is the statement in the tutorial an oversimplification and I'm being too pedantic?
If the codomain is the empty set and the domain is not, then the composition map cannot exist as there is no map from a non empty set into the empty set. Hence the existence of such a composition map rules out this situation.