Free Objects and Equational classes

Question

I can't give a proof of this fact: "Given a non-empty set $X$ and an equational class $V$, then $V$ contains a free object on $X$." $V$ is a class of algebras of a certain language of algebras $L$ and $V$, qua equational class, is $M(\Sigma)$ where $\Sigma$ is a set of identities and $M(\Sigma))$ indicates the class of $L$-algebras satisfying $\Sigma$. I wanna show that the couple $((W_{X},\Sigma),F)$ is an $L$-algebra, where $W_{X}$ is the set of words in the alphabet $X$ and with the only relations between words are given by $\Sigma$ and $F$ is the set of operations on $W_{X}$ given by the language $L$. The problem is: how to define $F$? Is this the right way to prove this fact? I don't know much about cathegory theory so I am trying this way.