Is there a sleek categorical description of the obvious functor $(\mathrm{Mono}\,\mathbf{Set})^{\mathrm{op}} \rightarrow */\mathbf{Set}$? Notation. Write $*/\mathbf{Set}$ for the category of pointed sets, and $F : \mathbf{Set} \rightarrow */\mathbf{Set}$ for the free functor (the "point-adjunction functor"). Given a category $\mathbf{C},$ write $\mathrm{Mono}\,\mathbf{C}$ for the wide subcategory of $\mathbf{C}$ whose arrows are the monomorphisms of $\mathbf{C}$.
The definition. There is an obvious functor $G : (\mathrm{Mono}\,\mathbf{Set})^{\mathrm{op}} \rightarrow */\mathbf{Set}.$ Explicitly, $G$ is defined as follows. Firstly, $G$ behaves just like $F$ when applied to an object. Secondly, given an injective function $m : X \rightarrow Y$, $G(m)$ is the pointed-set homomorphism $GY \rightarrow GX$ given as follows:


*

*If $y \in Y$ is in the image of $m$, then $G(m)(y)=m^{-1}(y)$.

*If $y \in Y$ is not in the image of $m$, then $G(m)(y) = *_X.$


Question. Is there a sleek categorical description of the functor $G$?
Motivation. Suppose $R$ is a ring. Consider a set of variables $X$, and a subset $S$ thereof. There is an obvious morphism $e_S : R[X] \rightarrow R[S]$ defined by taking every variable in $X\setminus S$ to $0$, and leaving the other variables untouched. One way to describe the morphism $e_S$ is as follows. There is a free functor $F_{R\mathbf{Alg}} : */\mathbf{Set} \rightarrow R\mathbf{Alg}.$ Now given $S \subseteq X$, the inclusion $m_S : S \rightarrow X$ gives rise to a "coinclusion" $G(m_S) : GX \rightarrow GS$. The morphism $e_S$ can then be described as $F_{\mathrm{R}\mathbf{Alg}}(G(m_S))$.
 A: To elaborate on Zhen Lin's comment, there is an object-fixing, contravariant involution on $\mathsf{Rel}$ given by taking the opposite relation. This restricts to an object-fixing, contravariant involution $I$ on the category of partial injections (This is an interesting fact about partial injections -- other interesting classes of relations such as functions or partial functions or injections or surjections are not closed under taking the opposite relation. Partial injections are exactly the relations $R$ such that for every $x$ there is at most one $y$ such that $xRy$ and at most one $z$ such that $zRx$. If a relation is viewed as a span $X \leftarrow R \rightarrow Y$ of functions, the partial injections are exactly those such that both legs of the span are injections).
Your functor is $G = H\circ I \circ J^{\mathrm{op}}$ where $J$ is the inclusion of injections into partial injections, $I$ is the involution on partial injections, and $H$ is the inclusion of partial injections into partial functions (or equivalently, pointed sets).
I'm not sure how "sleek" or "categorical" this account is -- it seems to me that category theory is only really being used as a language here, to describe some basic facts about sets. Each piece of the puzzle can presumably generalized to broader classes of categories, but personally I would tend to view that as an act of generalization rather than explanation.
