# A property of categories satisfied by categories of algebraic structures

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of $$B$$ into $$A$$, and for $$g_i$$ the mapping which is the same as $$f_i$$ as a function, but whose codomain has been replaced by $$B$$.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)?