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We investigate the (separated) monomorphism category$\operatorname{mono}(Q,\Lambda)$ of a quiver $Q$ over an Artin algebra$\Lambda$. We construct an epivalence from$\overline{\operatorname{mono}}(Q,\Lambda)$ to$\operatorname{rep}(Q,\overline{\operatorname{mod}}\, \Lambda)$, where$\operatorname{mod}\Lambda$ is the category of finitely generated modules and$\overline{\operatorname{mod}}\, \Lambda$ and$\overline{\operatorname{mono}}(Q,\Lambda)$ denote the respective injectivelystable categories. Furthermore, if $Q$ has at least one arrow, then we showthat this is an equivalence if and only if $\Lambda$ is hereditary. In general,it induces a bijection between indecomposable objects in$\operatorname{rep}(Q,\overline{\operatorname{mod}}\, \Lambda)$ andnon-injective indecomposable objects in $\operatorname{mono}(Q,\Lambda)$. Weshow that the generalized Mimo-construction, an explicit minimal rightapproximation into $\operatorname{mono}{(Q,\Lambda)}$, gives an inverse to thisbijection. Using this, we describe the indecomposables in the monomorphismcategory of a radical-square-zero Nakayama algebra, and give a bijectionbetween the indecomposables in the monomorphism category of two artinianuniserial rings of Loewy length $3$ with the same residue field.  These results are proved using free monads on an abelian category, in orderto avoid the technical combinatorics arising from quiver representations. Thesetup also specializes to representations of modulations. In particular, weobtain new results on the singularity category of the algebras $H$ which wereintroduced by Geiss, Leclerc, and Schr\"oer in order to extend their resultsrelating cluster algebras and Lusztig's semicanonical basis to symmetrizableCartan matrices. We also recover results on the $\iota$quivers algebras whichwere introduced by Lu and Wang to realize $\iota$quantum groups viasemi-derived Hall algebras.

A functorial approach to monomorphism categories II: Indecomposables