Let $C$ be a topos, $K$ a small category and $$ PSh(K) \leftrightarrows C $$ a reflective subcategory with inclusion $i\colon C\hookrightarrow PSh(K)$ and reflector $T$.
Is $T$ left exact?
$D$ being a topos is equivalent to the existence of a (probably different) reflective adjunction $$ PSh(K) \leftrightarrows D $$ with a left exact reflector.
No. Consider the functor $\Delta : \mathbf{Set} \to \mathbf{sSet}$ that sends each set $X$ to the discrete simplicial set on $X$. This is certainly fully faithful, and it has a left adjoint, namely the functor $\pi_0 : \mathbf{sSet} \to \mathbf{Set}$ that sends a simplicial set to its set of connected components. It is not hard to check that $\pi_0 : \mathbf{sSet} \to \mathbf{Set}$ does not preserve monomorphisms in general – so it does not preserve finite limits either.