TITLE: Examples of functors that preserves products but not equalizers, and vice versa.
QUESTION [6 upvotes]: What are simple examples, for student consumption, of 

A functor which preserves products (or at least finite products) but not equalizers.
A functor which preserves equalizers but not products.

Ideally, it would be good to have reasonably natural examples involving reasonably familiar categories and which don't call on esoteric background knowledge!

REPLY [6 votes]: Let $S$ be the category of sets (and functions), and let $F$ be the functor from $S$ to $S$ that sends the empty set to itself and sends every nonempty set to a singleton (and acts on maps in the only possible way).  This preserves products, but it fails to preserve the equalizer of the two maps from a singleton to a two-element set.
For the other direction, consider the functor $G:S\to S$ that sends objects $x$ to $x\times 2$ and sends maps $f$ to $f\times 1_2$ (where $2$ means a 2-element set and $1_2$ is its identity map).  This preserves equalizers but messes up products (even the empty product 1).