Motivation: I'm trying to figure out whether the image of an injective Lie group homomorphism (between simply connected Lie groups) under the Lie functor is injective. Since the Lie functor is an equivalence of categories from the category of simply connected Lie groups to the category of finite-dimensional real Lie algebras, one knows that the image of an injection under the Lie functor is a monomorphism in the category of finite-dimensional real Lie algebras.
Question: Are monomorphisms in the category of finite-dimensional Lie algebras over a field injective?
My thoughts: I know that monomorphisms in the category of all Lie algebras are injective, since the forgetful functor from the category of all Lie algebras to the category of sets has a left adjoint. However, the forgetful functor from the category of finite-dimensional Lie algebras to the category of sets doesn't seem to have a left adjoint. I'm also aware of the results that the Lie functor preserves surjections, and that there exists an epimorphism in the category of finite-dimensional Lie algebras that is not surjective, but I don't think that is useful for this question.
(Edit: Using that taking the kernel commutes with applying the Lie functor, I have been able to prove the question that motivated me: the Lie functor preserves injections.)
Best Answer
I claim that a morphism $f:\mathfrak{g}\rightarrow\mathfrak{h}$ is monic iff it is injective.
Of course, in any concrete category, injective implies monic, so we'll focus on the more fun direction.
So, assume $f$ is monic and let $\mathfrak{k}$ denote the kernel of $f$. Consider the two functions $g_1,g_2:\mathfrak{k}\rightarrow \mathfrak{g}$ where $g_1$ is the inclusion and $g_2$ is the zero map. Then $f\circ g_1 = f\circ g_2$ since both are the $0$-map. As $f$ is monic, we conclude that $g_1 = g_2$, that is, the that identity map of $\mathfrak{k}$ is the zero map. Thus, $\mathfrak{k} = \{0\}$, so $f$ is injective.