Terminology: Name for a homomorphism from the free object?

Is there a standard name for taking a homomorphism from the free object over an algebraic structure? Roughly speaking, this should amount to evaluation of any element of the free object under the homomorphism.

For example, take the free commutative monoid over the set of natural numbers; elements of this monoid are finite multisets of natural numbers. By taking commutative monoid homomorphisms we can express many common operations, such as cardinality (map every generator to 1 and take sum as the monoid operation), sum (take every generator to itself and map the monoid operation to sum of natural numbers), product (take product of natural numbers rather than sum) etc.

Since these operations are all abelian monoid operations, we can expect a sensible result which will not depend on our multiset being non-empty or listing its elements in a specific order.

Does this idea generalize to other algebraic structures? Any known references?

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The nLab has proposed the name "adjunct", which seems OK to me. So, for example, you could say something like "let $F(S)$ be the free group on the set $S$, let $G$ be a group, let $f:S\to G$ be a set map, and let $g:F(S)\to G$ denote the adjunct of $f$." I think it would be even a bit better to say "left adjunct" instead of just "adjunct".
Mac Lane used "left adjunct" in Theorem IV.1 (ii) ("...while each $g:x\to Ga$ has left adjunct [...] $Fx\to a$") –  user2734 Jan 25 '10 at 22:50