It is fun to recognize that the $n$-colorable graphs are exactly those graphs $X$ in the category of simple graphs with an homomorphism to the complete graph $K_n$.
Question 1: Are there other families of objects $Y$ in other categories - comparable with the complete graphs - such that the objects $X$ with a morphism to some $Y$ in this family are interesting enough to have a proper name (like "$n$-colorable graphs")? For example in the category of groups?
It seems natural to generalize the concept of colorability (in the first instance for graphs): Let a graph $G$ be $H$-colorable when there is a homomorphism from $G$ to $H$. The - arbitrary - graph $H$ defines which "colors" are allowed to be neighbours in a $H$-colored graph, and thus plays the role of a sort of a "grammar".
Question 2: Under which name has this generalization of colorability been investigated?
Question 3: Are there other categories - inhabited by a family of objects as in Question 1 - such that any target $Y$ can be understood as some kind of a "grammar" (as for graphs)?