The one-step generator extension functor

Let $\mathcal{A}$ be a locally small Grothendieck category with fixed generator $U$, and let $M$ be an object. Because subobjects of $U$ form a set up to equivalence and each hom-class is a set, one may form the set $S_M=\{(N,\phi )\mid N\subseteq U,\phi :N\to M\}\mathrm{.}$

For each $(N,\phi )\in S_M$, write $i_N:N\hookrightarrow U$ for the inclusion. The one-step generator extension of $M$ is the pushout $\begin{array}{ccc}{\displaystyle \underset{(N,\phi )\in S_M}{⨁}N}& ⟶& M\\ \downarrow & & \downarrow \\ {\displaystyle \underset{(N,\phi )\in S_M}{⨁}U}& ⟶& \mathbf{M}(M)\end{array}$ whose top map is induced by the $\phi$ and whose left map is induced by the $i_N$.

The canonical map $M\to \mathbf{M}(M)$ coming from the pushout is denoted ${\eta }_M$. A morphism $g:M\to M^{\mathrm{\prime }}$ carries each pair $(N,\phi )$ to $(N,g\circ \phi )$, so by the pushout universal property $M\mapsto \mathbf{M}(M)$ defines an endofunctor.