What is limit (singletone and arrow shape)?

Question

"L is a limit of diagram $F:\{*\} \rightarrow C$."

Does it mean that

"there exist $L\in C$ and map $f:L \rightarrow F(*)$ such that

for every $N\in C$ and map $g:N \rightarrow F(*)$

there exist unique map $u:N \rightarrow L$ such that

$f \circ u = g$ "?

It looks like expression "(L,f) is a terminal object in $C/(F(*))$ - slice category over $F(*)$".

Is it correct?

What is the limit of diagram which domain is arrow $\{A \rightarrow B\}$ ?

Answer 1

You are essentially correct, except that you must state that $u$ is unique in factoring $g$. That is, given $g\colon N\rightarrow F(\ast)$, there is a unique map $u\colon N\rightarrow L$ such that $g=u\circ f$. You can show that $L=F(\ast)$ and $f=1_{F(\ast)}$ is one choice of limit.

In general, it's a good exercise to show that if $F\colon I\rightarrow C$ is a diagram where $I$ has an initial object $i$, then one choice of limit for $F$ is $F(i)$ equipped with its identity map. For example, a limit of a diagram $F\colon \{A\rightarrow B\}\rightarrow C$ is $F(A)$ with its identity map.

I do not understand what you mean in this context by

"(L,f) is a terminal object in C/c - slice category over c".

What is $c$ here?