I know almost nothing about category theory, but I need some in my current work. I isolated a property of categories that I need to use in a proof, and that is shared by many classical categories, and I would like to know if it is well known, if it has a name, if it is implied by some classical property of categories...

Recall that a family of arrows $(f_i)$, all having the same codomain, is called collectively epimorphic if it satisfies the following condition: for every arrows $g, h$ whose domain is the same as the codomain of the $f_i$'s, if for every $i$, we have $g \circ f_i = h \circ f_i$, then $g = h$.

Say that a category $\mathcal{C}$ has property $(*)$ if for every family $(f_i)$ of arrows in $\mathcal{C}$, we can find a collectively epimorphic family $(g_i)$ on the same index set, and an arrow $h$, such that for every $i$, we have $f_i = h \circ g_i$.

If $\mathcal{C}$ admits coproducts, then $\mathcal{C}$ obviously satisfies $(*)$. But there are also natural categories that do not admit coproducts that satisfy $(*)$. For instance, consider the category $\mathcal{C}$ whose objects are a certain kind of algebraic structures (e.g. groups, rings, vector spaces over a fixed field...) and whose arrows are embeddings. Then $\mathcal{C}$ satisfies $(*)$. Indeed, if we denote by $A$ the codomain of the $f_i$'s, then we can let $B$ be the substructure of $A$ generated by the ranges of all the $f_i$'s, for $h$ the inclusion of $B$ into $A$, and for $g_i$ the mapping which is the same as $f_i$ as a function, but whose codomain has been replaced by $B$.

My question is the following:

Is there a classical property of categories that implies $(*)$, and that is satisfied by the kind of categories considered above (categories of algebraic structures with embeddings)?


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