This is exercise 2 in Mclane's Categories For the Working Mathematician, chapter 1.3.
Show that functors $1 \to C, 2 \to C, 3 \to C$ correspond respectively to objects, arrows and composable pairs of arrows in C.
(1 is the category of one object with just the identity morphism, 2 the category of two objects a,b with a morphism $a \to b$ and 3 the category with three objects $a$, $b$, $c$ with morphisms $a \to b,b \to c, a \to c$)
My question is perhaps very simple: When defining a functor T between two categories $C$, $D$ you define an object function from objects to objects and a morphism function. The object funtion is said to assign to each object $c$ an object $Tc$. So if $C$ has less objects than $D$ (as, for example happens in the category $1$), how does the functor work? In other words, the object funtion in the exercise from $1$ to $C$ where $C$ has more than one object, is not surjective, so there are objects in $C$ that are not represented by the functor.
I know there must be a mistake in my syllogism but can you clarify it a little, maybe with an example?