Exact functor on inclusion

Question

Let $\mathcal{C},\mathcal{D}$ be abelian categories. Let $F:\mathcal{C} \to \mathcal{D}$ be a covariant exact functor.

My questions:

1. How to show $F$ preserve monomorphism?

2. Let $\iota_A:A\to B$ be an inclusion. Does $F(\iota_A)=\iota_{F(A)}$, where $\iota_{F(A)}:F(A)\to F(B)$ is the inclusion on $F(A)$?

Answer 1

If $A \to B$ is a monomorphism, then you have an exact sequence $0 \to A \to B.$ Since $F$ is exact, you get the exactness of $0 \to F(A) \to F(B)$ which means that $F(A) \to F(B)$ is a monomorphism.