I am newbie to Category Theory and I just read about limit preserving functors and I was trying to prove that the functor $\mathscr{B} \rightarrow 1$ preserves limits. The book I am following is Leinster $\textit{Basic Category Theory}$ and it says the following
A functor $F: \mathscr{A} \rightarrow \mathscr{B}$ preserves limits iff for any diagram $D:I \rightarrow \mathscr{A}$ such that $\lim_ID$ exists, $\lim_IF\circ D$ exists and $\lim_IF\circ D$ =$F(\lim_I D)$.
So, in the context of the question I defined the diagram $D:I \rightarrow \mathscr{B}$ where the limit exists and is given by the following diagram
where $p_1,p_2$ are projections and $f$ is unique such that $p_1 \circ f=f_1$ and $p_2 \circ f=f_2$. Now, when I compose with $F$ I don't really know how to see that this limit, in fact, exists.
Does anyone have any intuition or a tip for me?
Any help is REALLY appreciated!