Typically in categories of algebraic objects (I will use $\mathsf{Grp}$ as an example), every object is the direct limit of its finitely generated subobjects. Based on this observation I was wondering how general this phenomenon is, and it turns out that the answer is locally finitely presentable categories.
The full definition is mostly very reasonable: A category $\mathcal C$ is called locally finitely presentable if the full subcategory $\mathcal C_{\mathrm{fp}}$ of compact objects is essentially small, it has small colimits, and every object is the filtered colimit of its compact subobjects.
Asking for $\mathcal C_{\mathrm{fp}}$ to be small makes sense, it's not very interesting that every object can be built of smaller objects if there's a crazy amount of small objects, asking for $\mathcal C$ to have colimits is also extremely reasonable, since we're trying to build everything as a colimit, and focussing on filtered colimits is a sensible generalization from the direct limits of the algebraic categories.
What is not clear to me is why do we use compact objects, where $c\in\mathcal C$ is called compact if the functor $\mathcal C(c,-)$ preserves filtered colimits. Why is preserving filtered colimits a smallness condition on objects? And why is compact objects the "correct" notion of small objects to consider?
My naive approach to generalize this phenomenon from algebraic categories would have been to move to a concrete category $\mathcal C$ with free objects (meaning that the forgetful functor $\mathcal C\to\mathsf{Set}$ has a left adjoint $F$) and call an object $c\in\mathcal C$ finitely generated if there is some finite $x\in\mathsf{Set}$ such that there is an epimorphism $F(x)\to c$. Let's say that a category $\mathcal C$ satisfies property (N) (for naive and lack of a better name) if $\mathcal C$ is concrete with free objects and every $c\in \mathcal C$ is the filtered colimit of its finitely generated subobjects.
Suppose that $\mathcal C$ is such that both property (N) and being finitely locally presentable make sense for $\mathcal C$ (so $\mathcal C$ is concrete with free objects, an essentially small subcategory of compact objects and has all small colimits), is it true that $\mathcal C$ has property (N) iff it is locally finitely presentable? (Note that even in $\mathsf{Grp}$ compact and finitely generated objects are not the same).