*The slice category or over category $C/c$ of a category $ C $ over an object $ c∈C $ has
objects that are all arrows $ f∈C $ such that $ cod(f)=c $,
and
morphisms $ g:X→X'∈C $ from $ f:X→c $ to $ f':X'→c $ such that $ f'∘g=f $.
*There is a forgetful functor $ U_{c}: C/c→C $ which maps an object $ f: X→c$ to its domain $ X $ and a morphism $ g:X→X'∈C/c $ (from $f:X→c$ to $f':X'→c$ such that $f'∘g=f$) to the morphism $g: X→X'$.
Please i need to know how one can define free objects using the free functor in that category.
The forgetful functor does not have a left adjoint in general, because the identity on $c$ is the terminal object of $C/c$, even if $U_c(id_c)=c$ is not a terminal object in $C$. In fact, if $c$ is terminal then $U_c$ is an isomorphism, so $U_c$ has a left adjoint if and only if it has an inverse.
On the other hand, if $C$ has products then $U_c$ has a right adjoint, which takes an object $X$ to the projection $X\times c\to c$.