Let $\mathcal{C}$ be a concrete category and $X$ be a set, let $F_X$ be free on $X$ with canonical injection $i:X\to F_X$. Is it always true that $i(X)$ generates $F_X$?
It looks like an easy statement to prove, but I have not found a nice categorical argument to show this.
Let $A$ be a subobject of $F_X$ such that $i(X)\subseteq A$. By the universal property of free objects, there exists exactly one morphism $\phi\colon F_X\to A$ such that $\phi|_{i(X)}$ is the identity map. Then $\operatorname{inc_A^{F_X}}\circ \phi\colon F_X\to F_X$ and $\operatorname{id}_{F_X}\colon F_X\to F_X$ are two morphisms that are the identity map on $i(X)$. Hence again by the universal property, $\operatorname{inc_A^{F_X}}\circ \phi=\operatorname{id}_{F_X}$. Every $x\in F_X$ has pre-image $\phi(x)$ in $A$, thus the inclusion $\operatorname{inc_A^{F_X}}$ is surjective and $A = F_X$.
$$\begin{matrix}&&X\\ &\stackrel i\swarrow&\downarrow&\stackrel i\searrow\\ F_X&\stackrel\phi\rightarrow& A&\hookrightarrow &F_X\end{matrix} $$