I've been reading the book Handbook of categorical algebra, by F. Borceux, and I'm a bit stuck with this result.
'Proposition. Consider a category $\mathcal{C}$ and a fixed object $I \in \mathcal{C}$. If $\mathcal{C}$ is complete, $\mathcal{C}/I$ is complete.
Proof. Consider a non-empty family of objects $(f_k : C_k \longrightarrow I)_{k \in J}$ in the category $\mathcal{C}/I$. In $\mathcal{C}$, the diagram constituted by all those morphisms $f_k$ has a limit given by an object $L$ and morphisms $p_k : L \longrightarrow C_k$, $p : L \longrightarrow I$.'
Then the proof goes on, but I am a little confused here. First, what does the author mean by 'diagram' here and why does it have a limit? I assume you can think of a functor coming from another category to use the completeness of $\mathcal{C}$. Second, how does he construct the arrow $p : L \longrightarrow I$?
Thank you
The diagram is a functor from a category $J\cup\{\star\}$ where $\star\notin J$, which has arrows $k\to \star$ for all $k\in J$ (and identity arrows of course), and that sends $k\to C_k$, $\star\to I$, and the unique arrow $k\to \star$ to $f_k : C_k\to I$.
This is a diagram and by definition of completeness it has a limit $L$ in $\mathcal{C}$, denoted by $L$ which consists of a family of arrows $p_k : L\to C_k$ and an arrow $p: L\to I$ such that all the involved diagrams commute (that is $f_k\circ p_k = f_j\circ p_j = p$ for all $k,j\in J$)
(You can think of it like a "giant pullback" with more than two objects)