Let $G$ be an abelian group. Let $\widehat{G}$ be a profinite completion of $G$. Profinite completion means a inverse limit of $G$ by a system given by homomorphisms $G/N\to G/M$ where $N$ and $M$ are finite index subgroups of $G$ and $N\subset M$.
Let $G_1$ and $G_2$ be groups and suppose there is an injection $G_1\to G_2$.
Is it true that $\widehat{G_1}\to \widehat{G_2}$ also injective ?
I know that inverse limit is a left exact functor. But when it comes to profinite completion, I don't have a confident.$G/N\to G/M$ is not always injective, so I cannot use the left exactness of inverse limit.