I am having a hard time proving the truth of this exercise I ran into:
If $\mathcal{C}$ is a small category with finite limits and $F: \mathcal{C} \to \mathbf{Set}$ is a functor, then $F$ preserves finite limits if and only if for any set $S$, $(S \downarrow F)$ has finite limits.
Anyone know a proof of this?
Consider a finite diagram $J: I\to \mathcal{C}$, let $L$ be its limit with projections $p_i: L\to J(i)$.
Let $X$ be a set with a cone $f_i : X\to FJ(i)$.
Consider the category $(X\downarrow F)$. By hypothesis it has a limit for the diagram we just obtained (the $f_i$ + the maps between the $FJ(i)$'s obtained from $J$), say $f: X\to F(A)$ (naming the projections is not interesting here)
But this means we have a cone $m_i: A\to J(i)$, so it factors uniquely through $L$, say with $g: A\to L$; and so we get $F(g)\circ f : X\to F(L)$ making everything commute.
Moreover, this map is unique, indeed if this map $g: X\to F(L)$ exists, then by the limit property of $f: X\to F(A)$ it must factor (in $(X\downarrow F)$) through $f$ and so it must be written $F(g)\circ f$ for some $g$; and then the $g$ must be the one we get from the limit property of $L$.
This proves that the cone $F(L), F(p_i)$ is a limit cone in $Set$ and so $F$ preserves finite limits.
Conversely, if $F$ preserves finite limits then clearly for every $X$, $(X\downarrow F)$ has finite limits that are given by the limits in $\mathcal{C}$.