Morphisms in the category of group presentations What are the morphisms in the category of group presentations?
 A: I've never heard of the "category of group presentations". If I had to guess what was intended by such a name, it would be a category whose objects are diagrams
$$ F \to G $$
where $F$ and $G$ are free groups. Correspondingly, the morphisms ought to be morphisms of diagrams: that is, commutative squares
$$ \begin{matrix} F &\to& G \\ \downarrow & & \downarrow \\ F' &\to& G' \end{matrix} $$
A: If you're happy for the objects of the resulting category to be quite "rigid" (i.e. being a morphism is a very strong condition), then the following definitions get the job done.

Definition 0. Whenever $S$ is a set, a quiver structure on $S$ consists of a set $Q$ together with two maps $$\mathrm{tar}_Q : S \leftarrow Q, \qquad \mathrm{src}_Q : S \leftarrow Q.$$

We think of $Q$ as a family of arrows between the elements of $S$; given $q \in Q$, we think of $\mathrm{tar}_Q(q)$ as the "target" of $q$ and $\mathrm{src}_Q(q)$ as the "source."

Definition 1. A group presentation $P$ consists of a set $\mathrm{gen}(P)$ together with a quiver structure $\mathrm{rel}(P)$ on the underlying set of the group freely generated by $\mathrm{gen}(P).$

For example, let $P$ denote the presentation $\langle x,y \mid x^2=y^3\rangle.$ Then $P$ can be described abstractly as follows:


*

*$\mathrm{gen}(P) = \{x,y\}$

*$\mathrm{rel}(P)$ is a set with one element, call that element $*$

*$\mathrm{tar}_{\mathrm{rel}(P)}(*) = x^2$

*$\mathrm{src}_{\mathrm{rel}(P)}(*) = y^3$


We think of the elements of $\mathrm{gen}(P)$ as our generators and the elements of $\mathrm{rel}(P)$ as our relations.
Now write $F$ for the free group functor.

Definition 2. Given group presentations $Q$ and $P$, a morphism $f : Q \leftarrow P$ consists of two functions $$\mathrm{gen}(f) : \mathrm{gen}(Q) \leftarrow \mathrm{gen}(P), \qquad \mathrm{rel}(f) : \mathrm{rel}(Q) \leftarrow \mathrm{rel}(P)$$ such that the following equations hold:
$$F(\mathrm{gen}(f)) \circ \mathrm{tar}_{\mathrm{rel}(P)} = \mathrm{tar}_{\mathrm{rel}(Q)} \circ \mathrm{rel}(f), \qquad F(\mathrm{gen}(f)) \circ \mathrm{src}_{\mathrm{rel}(P)} = \mathrm{src}_{\mathrm{rel}(Q)} \circ \mathrm{rel}(f)$$

