Is there difference between "preserving equalizers" and "preserving regular monomorphisms"? I know that an arrow $e:a\to b$ in some category $\mathcal A$ is by definition a regular monomorphism if an object $c$ exists together with arrows $u,v:b\to c$ such that $e$ serves as equalizer of $u$ and $v$.
If also $\mathcal B$ is a category and $F:\mathcal A\to\mathcal B$ is a functor that preserves equalizers then - if I understood well - we are allowed to conclude that $Fe$ serves as equalizer of $Fu$ and $Fv$.
But what if $F$ is (only) said to preserve regular monomorphisms?
Is that exactly the same information about $F$ or is it less information because we are not allowed to conclude that regular monomorphism $Fe$ serves as equalizer of $Fu$ and $Fv$?
The best scenario is of course that it can be proved that in this situation $Fe$ is an equalizer of $Fu$ and $Fv$, but it seems to me that extra conditions are needed for that (or am I wrong here?).
 A: As I said in the comments, in Abstract and Concrete Categories- The Joy of Cats one can read the following statement, page 116 (in the 2004 edition) "we will see that it is possible for a functor to preserve regular monomorphisms without preserving equalizers (13.6)". 
A way to find such a counterexample is to pick a category $\mathcal{B}$ that has many regular monos, for instance one where all monos are regular ($\mathbf{Ab}, Set$, any topos,...) and a functor that preserves monos but not equalizers.
The (ad hoc) example I gave is the following: pick any such category $\mathcal{B}$ (for instance $Set$) , and three arrows $A\to B$, $B\to C, B\to C$ such that the arrow $A\to B$ equalizes the two other arrows, but is not an equalizer of them (and is a monomorphism).
Then pick $\mathcal{A}$ to be the diagram of an equalizer (that is a category with objects $1,2,3$ and arrows that compose in the obvious way- see my comment for more detail), and choose the obvious functor $F: \mathcal{A}\to \mathcal{B}$ sending $1\to A, 2\to B, 3\to C$. 
Clearly this functor is a counterexample to "a functor preserving regular monos preserves equalizers"
