Let $C$ be a category. Roughly, a model of a (finite) limit sketch in $C$ is a functor $S \to C$ where $S$ small category with some specified (finite) cones which are sent to limits in $C$. $S$ itself is called a sketch. (The nLab page is a bit brief. There is a lot more in chapter D2 "Sketches" in the Elephant )

An example is the sketch for categories, which has four objects $X_0,X_1,X_2,X_3$ (representing objects, arrows, composable pairs and composable triples resp.), arrows representing the various structural maps in the definition of a category (identity arrow, source, target, composition etc and their compositions) and equations between arrows representing the usual equations in the definition of a category. There are cones which tell us that $X_2$ is the usual pullback of the source and target maps, and similarly for $X_3$. These cones are mapped to actual pullbacks in $C$.

My question is, what if we want to not just map cones to limits, but map certain arrows in $S$ to specified classes of arrows in $C$? Has this been considered in the literature at all?

One simple example might be mapping certain maps to monomorphisms. When monomorphisms are expressible by universal properties then this seems to be encapsulated by the existing definition. Or if we mix things up and allow more general sketches, with colimits as well as limits, then maps such as regular epimorphisms could be singled out by the definition. But I could imagine more general examples.

And the more interesting question is this: given such an extension to the concept of sketch, what can we say about the category of models of such sketches? In this MO answer we see that accessible categories are characterised up to equivalence as being models of a sketch. Does such a classification seem possible for this extended concept?

alwaysexpressible via universal properties: in any category, $f : X \to Y$ is monic if and only if the following diagram is a pullback square: $$ \begin{matrix} X & \xrightarrow{1_X} & X \\ \downarrow 1_X\!\!\!\!\!\!\!\! & & \downarrow f \!\!\!\! \\ X & \xrightarrow{f} & Y \end{matrix} $$ – Hurkyl Nov 18 at 13:01