Elementary question about morphisms in Category Theory I'm getting into a tangle over absolutely elementary definitions in Category theory.
Consider an arbitrary category with some number $n$ of objects (say $n > 2$). Pick two objects in the Category, say $A$ and $B$. Does there (by definition) exist a morphism between $A$ and $B$?
I'd always thought the answer was yes (until I actually started studying some Category theory in earnest - today). I now think the answer is no.
 A: Not necessarily.  Here's an easy example. Take the category of sets.  If $A$ is not empty, then there can be no morphism $A \to \varnothing$. 
Other easy (and more severe) examples abound.  Take objects to be sets again, but now let $\mathrm{Hom}(A,B)$ be the set of all surjections $A \to B$. Again, this is a category, but it is "lacking" lots of morphisms.  Of course, you can play the same game with injective maps. 
A: No, the only morphisms that need to exist are the identity morphisms. So e.g. a two-object category with only identity morphisms (in particular, no morphism from one object to the other) is fine.
A: You're correct they are not required to exist. I find the easiest example to be an ordered sets like $\mathbb{N}$ with $\mathrm{Hom}(A,B)$ being defined by $\leq$. In this case for $A \neq B$ the existence of $\mathrm{Hom}(A,B)$ strictly forbids the existence of $\mathrm{Hom}(B,A)$.
A: There are many examples of categories that have this property. Any category with a zero object $z$, such as the category of abelian groups, has this property, because for any two objects $X \to Y$, there always exists the zero map between them, namely the composite $X \to z \to Y$.
Conversely, there are many counterexamples. Any category that has a terminal object $t$ and an initial object $i$ that are not isomorphic, such as the category of sets, then any object $X$ can satisfy at most one of the following two properties:


*

*There exists a morphism $t \to X$

*There exists a morphism $X \to i$.


In particular at least one of $\hom(t, X)$ and $\hom(X, i)$ must be empty.
