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?