Consider the direct image functor $f\_\*: Sh(X) \rightarrow Sh(Y)$, let $X$ and $Y$ be topological spaces, let $f: X \rightarrow Y$ be a continuous map, let $G \in Sh(X)$ be a sheaf. I was reading this course on sheaves:


<http://bit.ly/14IBBTZ>.


On page 1 they define this subpresheaf $f\_!G \subset f\_\*G$ then on page 2 in the first line they remark that: 


"As $f\_!$ is a subfunctor of $f\_\*$ it is left exact".


I wanted to ask, are subfunctors of left exact functors also left exact?



Are subfunctors of left exact functors also left exact?
Here's a counterexample with additive functors on abelian categories. If $A$ is an abelian group, let $F(A)$ denote the subgroup of elements that are divisible by $2$. It is easy to see that $F:Ab\to Ab$ is an additive functor, and $F$ is a subfunctor of the identity. But $F$ is not left exact because it does not preserve kernels. For instance, $F$ sends the exact sequence $$0\to \mathbb{Z}\to\mathbb{Q}\to\mathbb{Q}/\mathbb{Z}$$ to $$0\to2\mathbb{Z}\to\mathbb{Q}\to\mathbb{Q}/\mathbb{Z},$$ which is not exact.


