Does the limit of a diagram with a single arrow exist?

Question

Sorry if it’s a pointless question! I’m trying to self-learn category theory (not easy), but none of the books i’ve looked at explains this.

When they introduce limits, they give the definitions in terms of diagrams and cones and then all of them offer the same 3 examples: the limit of the empty diagram is the terminal object, the limit of a diagram with no (nontrivial) morphisms is the product, and the limit of a diagram with two points and two parallel arrows is the equalizer. The obvious question is how about the limit of a diagram with a single (nontrivial) arrow? I can think of several answers:

• It can‘t exist because of some theorem
• It can exist, but it has no interesting properties

I’ve tried to think about it but I'm not able to tell.. Any answer and/or resource on this would be really appreciated!

Ps: it is somehow related to the exponential/internal hom object? The definitions look different of course, but maybe they are partially related?

Answer 1

If you're thinking about the diagram $\bullet \to \bullet$ (and identity arrows), then its limits are indeed uninteresting.

In fact, the limit of $A\to B$ is always just $A$.

The more general statement here is :

Suppose $I$ is a category with an initial object $c$. Then any functor $F:I\to C$ has a limit, and it's given by $F(c)$.

(actually there's an even more general statement about cofinality and related things, but let's stick to that for now)

This statement is pretty easy to prove, and you should try it for yourself. Then, notice that the first dot in $\bullet \to \bullet$ is initial.