I am aware that the following question is a very basic one and therefore I would not be at all offended if it were to be closed. Moreover, I am not familiar at all with category theory.


Let $\mathcal{C}$ be a concrete category and $X$ be a free object of $\mathcal{C}$. 



> 
> If $Y\_1$ and $Y\_2$ are both free subobjects of $X$, then is the
>  intersection, $Y\_1 \cap Y\_2,$ free?
> 
> 
> 



Intersection of free objects
It's not in general true for the category of modules for a ring.


For example, let $R=\mathbb{C}[x]/(x^2)$, let $X=R\oplus R$ be the free module on two generators, and let $Y\_1$ and $Y\_2$ be the submodules of $X$ generated by $(1,0)$ and $(1,x)$ respectively. Then $Y\_1$ and $Y\_2$ are both free modules on one generator, but $Y\_1\cap Y\_2$ is one-dimensional, spanned by $(x,0)$, and is not free.


