If $x$ is an object of a category $C$, one usually says that $x$ is if finite presentation (or compact) if for any direct filtered system ($y_i$) in $C$, the canonical map

$$f : \text{colim}_i \, \text{Hom}(x,y_i) \to \text{Hom}(x, \text{colim} y_i)$$

is bijective. One usually says $x$ is of finite type if this holds only for direct filtered systems of monomorphisms.

However, I am interested in objects $x$ for which $f$ is injective (without additional condition on the direct filtered system). It seems to me that, in the category of (right) modules over a ring, such objects are exactly finitely generated modules. My question is, has this been considered, and is there a name for this property of $x$? I considered "finitely generated"... but does anyone have a better idea?