# Why is the category $Elts(F)$ complete and the forgetful functor $\phi_F:Elts(F)\rightarrow \mathscr{C}$ limit preserving?

Borceux - Handbook of categorical algebra p.91

Proposition 2.16.1 Let $\mathscr{A},\mathscr{B}$ be complete categories, and $F:\mathscr{A}\rightarrow\mathscr{C}$, $G:\mathscr{B}\rightarrow\mathscr{C}$ be limit preserving functors. Then, the comma category $F\downarrow G$ is complete and the projection functors $U:F\downarrow G\rightarrow \mathscr{A}$, $V:F\downarrow G\rightarrow \mathscr{B}$ are limit preserving.

There is a corollary right after this proposition.

Corollary 2.16.2 Let $\mathscr{C}$ be a complete category and $F:\mathscr{C}\rightarrow \textbf{Set}$ be a limit preserving functor. Then, the category $Elts(F)$ is complete and the forgetful functor $\phi_F:Elts(F)\rightarrow \mathscr{C}$ is limit preserving.

I don't understand how this corollary can be derived from the above proposition. Let $1:\textbf{1}\rightarrow \textbf{Set}$ be a functor (a functor which selects a singleton). It is written in the text that by applying the proposition to the pair $(1,F)$, we get the corollary. However, to apply the proposition, $1$ must be limit preserving, but I don't think this is limit preserving.

Let $\mathscr{D}$ be the discrete two-points category and $G:\mathscr{D}\rightarrow \textbf{1}$ be a functor, and let's denote $\textbf{1}=\{\ast\}$. Obviosly, $\ast$ is the limit of $G$, but $1(\ast)$ is not necessarily the limit of $1 \circ G$. Indeed the limit is $1(\ast)\times 1(\ast)$.

So, how can one derive the corollary from the proposition? Thank you in advance.

## 1 Answer

"Limit-preserving" means up to the canonical isomorphism, and the product of two singletons is a singleton. In general, the functor picking out a terminal object preserves all limits, because it's a right adjoint.

• Aha right. Singleton is terminal! Thank you. – Rubertos Feb 18 '17 at 8:41