Always.

In the case of algebraic theories, the proof is simple. Let $F : Set \to V$ be the free functor, where $V$ is some algebraic theory. Then if $X$ is a set and $U$ a $V$-algebra,

$$ \operatorname{Hom}_{V}(F(X),U) \cong \operatorname{Hom}_{\operatorname{Set}}(X,|U|) $$

The right-hand side is a $V$-algebra, naturally in both $X$ and $U$. Therefore $F(X)$ is a co-$V$-algebra object in $V$, which is more structured than just being a $V$-algebra. Furthermore, if $X$ is a monoid then $F(X)$ becomes a Tall-Wraith $V$-monoid (obligatory n-lab reference: Tall-Wraith monoids).

(The details of this particular argument are a part of a forthcoming paper by Sarah Whitehouse and myself on Tall-Wraith monoids.)

Always.

In the case of algebraic theories, the proof is simple. Let $F : Set \to V$ be the free functor, where $V$ is some algebraic theory. Then if $X$ is a set and $U$ a $V$-algebra,

$$ \operatorname{Hom}_{V}(F(X),U) \cong \operatorname{Hom}_{\operatorname{Set}}(X,|U|) $$

The right-hand side is a $V$-algebra, naturally in both $X$ and $U$. Therefore $F(X)$ is a co-$V$-algebra object in $V$, which is more structured than just being a $V$-algebra. Furthermore, if $X$ is a monoid then $F(X)$ becomes a Tall-Wraith $V$-monoid (obligatory n-lab reference: Tall-Wraith monoids).

(The details of this particular argument are a part of a paper by Sarah Whitehouse and myself on Tall-Wraith monoids.)