A functor which is limiting-object-preserving but not limit-preserving Consider the following "proposition":

"Proposition." Let $A$, $B$, $C$ be categories, such that $B$ and $C$ have all limits from $A$; $T\colon B\to C$ be a functor. Then $T$ preserves all limits from $A$ iff $T$ preserves all limiting objects from $A$.

Questions:
1). Can anyone give a counterexample to this fake proposition?
2). Is there some positive results (probably with some assumptions on $B$ and $C$), making this "proposition" a proposition?
 A: Here's a simple counterexample.  Take your favorite category $C$ with binary products and an object $X$ such that $X\times X\cong X$ and the two projection maps $X\times X\to X$ are not isomorphisms (e.g., any infinite set in $\mathtt{Set}$).  Then for any category $B$ with binary products, the constant functor $B\to C$ with constant value $X$ preserves binary product objects, but not binary product diagrams.  (More generally, any functor $B\to C$ that sends every object to $X$ will preserve binary product objects, but most probably won't preserve binary product diagrams.)
In particular, in this counterexample $B$ and $C$ can be as nice as you could imagine (e.g., they could both be $\mathtt{Set}$, or pretty much any other familiar concrete category).  So I doubt there are any reasonable conditions on $B$ and $C$ alone which can give a positive answer; you need to also put some restriction on $T$ (and/or maybe on $A$: for instance, if $A$ is empty, then trivially preserving limit objects implies preserving limit diagrams).
