Let $\textbf{Sh}(X)$ be the category of sheaves of sets on a topological space $X$. A continuous map $f:X \to Y$ induces a pair of adjoint functors $f^{-1}:\textbf{Sh}(Y) \to \textbf{Sh}(X)$ and $f_*:\textbf{Sh}(X) \to \textbf{Sh}(Y)$. Why does $f^{-1}$ preserve finite limits?
I'm having trouble constructing the necessary unique maps from $\mathcal{E} \to f^{-1}\mathcal{L}$ for a finite limit $\mathcal{L} \in \textbf{Sh}(Y)$. Adjunction doesn't seem to help since my arrows are going in the wrong direction.
I can get an arrow $f^{-1}f_*\mathcal{E} \to f^{-1}\mathcal{L}$ by adjunction, but I only get a natural map $f^{-1}f_*\mathcal{E} \to \mathcal{E}$.
There is a really explicit description of $f^{-1}$ as $$ f^{-1}{\cal G}:U\mapsto \underset{V\supseteq fU}{\text{colim }} {\cal G}V $$ ($\cal G$ a sheaf on $Y$, $U$ an open subset of $X$, $V$ open subsets of $Y$); this colimit is filtered, and since (a) finite limits in ${\bf Sh}(A)$ are computed pointwise and (b) filtered colimits commute -in $\bf Set$- with finite limits, you have the result that $f^{-1}$ is left exact.
There are, I think, even more formal ways to prove this statement. Whether you want them or not depends on what grip you want on the result :-)