I recently started studying category theory, and all the books give roughly the same definition of a category:
A category $C$ consists of class of objects $Ob_C$. For each two objects $A, B \in Ob_C$ there is a set (class) of morphisms $Hom(A, B)$, ... etc.
However none of these books says that $Hom(A, B)$ can't be empty (however, it is not claimed that it can be empty).
So, I have a question if $Hom(A, B)$ can be empty (except in those cases when it must be not empty because of the composition rule). I would be very grateful if someone would give a source where this is clearly stated.
Also if $Hom(A, B)$ can be empty, than what is the name for categories in which all the objects are connected (something like fully connected category probably)?