another naming question from me, which comes up because I try to use category theory as a compass in developing some (new or not?) order-theoretic notions.
According to my book (The Joy of Cats, [AdamekHerrlichSchrecker]), in a concrete category, an object $F$ is co-free over a set $X$, with canonical function $f: |F| \rightarrow X$, if for every other object $G$ and function $g : |G| \rightarrow X$ there is a unique morphism $u : G \rightarrow F$ such that $f \cdot |u| = g$ (with |.| denoting the forgetfull functor).
Now, in my own favourite concrete category, co-free objects are not always there, but for every set $X$ I am able to create a (for me meaningful) object $F$ with the property that for every injection $g : |G| \rightarrow X$ there is a unique morphism $u : G \rightarrow F$ such that $f \cdot |u| = g$.
What I would like to know is whether there is a name in literature for this property as well?
Concrete example: my favourite category is the category of 'prefix orders' or 'generalized trees', i.e. the category of partial orders in which every downward closed set $x^- = \{ y \mid y \leq x \}$ is totally ordered. Morphisms in this category are partial history preserving maps, i.e. partial functions $f : X \rightarrow Y$ such that $f(x^-) = f(x)^-$ for all $x$. In this category, for a given set X, the set $X^*$ of all total orders over subsets of X that have a maximum can be considered as a tree (using the natural prefix ordering on those total orders). This tree $X^*$ has the property I suggest, but is not co-free. I would like to call it the 'free tree' over $X$ first, but then found out it is actually more (but not exactly) like a 'co-free tree'. The canonical function in this case is max() of course.