Free functor

Remarks

Such a functor exists for every variety: $F(X)$ is the free algebra of $\mathcal{V}$ on $X$, and $F$ sends a map $h \colon X \to Y$ to the unique homomorphism $F(X) \to F(Y)$ that extends $\eta_Y \circ h$. It is unique up to a unique natural isomorphism, as is any left adjoint. More generally, a left adjoint of any forgetful functor $U \colon \mathcal{C} \to \mathcal{D}$ is called a free functor.

Example: for the variety of monoids, $F(X)$ is the monoid $X^*$ of finite words in $X$ under concatenation, and $\eta_X$ sends $x$ to the one-letter word $x$. A map $g \colon X \to M$ into a monoid extends to $\bar g(x_1 \cdots x_n) = g(x_1) \cdots g(x_n)$, and this is the only homomorphism extending $g$.

Non-example: the functor from sets to groups that sends every set to the trivial group is not a free functor. For $X = \set{x}$ and the map $x \mapsto 1$ into $\Z$, the only homomorphism from the trivial group to $\Z$ sends everything to $0$, so it does not extend $g$.