Examples of functors that preserves products but not equalizers, and vice versa. What are simple examples, for student consumption, of 


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*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!
 A: 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).
A: Let $R$ be a ring and let $M$ be a left $R$-module. Tensoring with $M$ gives a cocontinuous functor $(-) \otimes_R M$ from right $R$-modules to abelian groups. It preserves equalizers iff $M$ is flat and it preserves products iff $M$ is finitely presented, so


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*Find a module which is finitely presented but not flat. For example, we can take $R = \mathbb{Z}, M = \mathbb{Z}_2$. The corresponding functor takes an abelian group $A$ to the tensor product $A \otimes \mathbb{Z}_2$. This preserves products but it does not preserve the equalizer of the diagram $\mathbb{Z} \xrightarrow{0, 2} \mathbb{Z}$. 

*Find a module which is flat but not finitely presented. For example, we can take $R = \mathbb{Z}, M = \mathbb{Q}$. The corresponding functor takes an abelian group $A$ to the tensor product $A \otimes \mathbb{Q}$. This preserves equalizers but it does not preserve the product of countably many copies of $\mathbb{Z}$. 
