Pullback preserving Set-functors

[Tobias Schroeder <tschroed@Mathematik.Uni-Marburg.de>]

I'm interested in proofs of or counterexamples for
the following conjectures:

Conjecture 1:
Any Set-endofunctor that preserves
kernels (i.e. pullbacks of a mapping with itself)
preserves pullbacks.

Conjecture 2:
Any Set-endofunctor that preserves
kernels *and inverse images* (i.e. pullbacks
where one of the mapping is injective)
preserves pullbacks.

Conjecture 3:
Same as Conj. 2 with *and inverse images*
replaced by *and equalizers*.

Conjecture 4:
Same as Conj. 1-3, but concerning *weak*
preservation.

Conjectur 5:
Same as 1-4, but for Set-endofunctors
that are subfunctors of a pullback
preserving functor.

I tried to prove these facts in several ways
but was not able to do it or to find a
counterexample (the answer to this questions
is of some relevance for my work on coalgebras)
... and it looks quite easy, doesn't it?

Can somebody help me in this?

Thank you very much in advance

Tobias Schröder
--------------------------------------------------------------
Tobias Schröder
FB Mathematik und Informatik
Philipps-Universität Marburg
WWW: http://www.mathematik.uni-marburg.de/~tschroed
email: tschroed@mathematik.uni-marburg.de