What means an empty diagram? An empty category $0$ is the category without objects and morphisms. A diagram with empty category as source called empty diagram, $F:0\rightarrow C$. But what means this exactly??
 A: This is the same thing, basically, as a function from the emptyset to another set. A function from $X$ to $Y$ is a subset of $X\times Y$ with certain properties; it's a good exercise to show that if $X=\emptyset$, then $\emptyset$ is a function from $X$ to $Y$ (the only one, in fact). This relies on the vacuousness of quantifying over the emptyset. Any statement of the form, "Every element of the emptyset is [blah]" is automatically true.
Identically, there is a unique functor from $0$ to $C$ for any category $C$; it's a good exercise, following the previous exercise, to prove this statement.
Now to get a feel for how to reason about empty sets/categories, let's compute the limit of the empty diagram in $C$ (if it exists). The limit of a diagram $\mathcal{D}$ is an object $o$ together with arrows $\alpha_d: o\rightarrow d$ for each object $d$ in $\mathcal{D}$, such that

*

*the resulting larger diagram commutes (for each arrow $\beta: d\rightarrow d'$ in $\mathcal{D}$, we have $\beta\circ \alpha_d=\alpha_{d'}$); and


*$(o, \{\alpha_d: d\in\mathcal{D}\})$ is universal with the above property: whenever $(s, \{\gamma_d: d\in\mathcal{D}\})$ satisfies the above property as well, there is a unique arrow $c: o\rightarrow s$ such that for each $d\in\mathcal{D}$ we have $\alpha_d\circ c=\beta_d$.
Alright, so let's look at the empty diagram. The limit is going to consist of an object $o$, together with a set of arrows. How many arrows? Well, none! This automatically means that any object satisfies the first bullet above since that bullet is a universal quantification over the emptyset.
Now let's look at the second bullet. Again, because of the universal quantification over the emptyset, the second bullet amounts to

*

*there is a unique arrow from $s$ to $o$, for any object $s$.

That is, $o$ is the terminal object of $C$! So we have:

The limit of the empty diagram in $C$ is just the terminal object of $C$, if it exists; and if $C$ has no terminal object, then there is no limit of the empty diagram in $C$. (Remember that in arbitrary categories, arbitrary diagrams need not always have limits!)


Note that above I kept using the word "the": "the" limit, "the" initial object, etc. This isn't actually good practice - really, each of these notions is only defined up to isomorphism (and this is true of most notions in category theory - indeed, non-isomorphism-invariant notions are often considered evil!). So throw the words "up to isomorphism" into the answer above, a whole bunch of times, if this is something you want to be explicit about.
