Diagram is from some book on categories shelf (ID?)- It's a good visual to rembember. What are some obvious extensions of categorical properties? Ie valid in every category, eg:
strong-, extremal-, regular-, split- ...
Are injection and surjection even categorical? Is the assumed context in n-lab clear?
EDIT:
I rephrased the title to hopefully clarify (didn't want to confuse "extending" and "diagram" with, say "linear extensions" and categorical diagrams).
Also, although it's valuable to see injection and surjections in context, the question focuses on categorical properties.
Injection and surjection can be usefully defined at least in concrete categories. You can say that a morphism $f$ in a concrete category $(\mathcal C, U)$ is injective if $Uf$ is an injection. As the forgetful functor $U$ is assumed to be faithful, it in particular reflects monomorphisms, so $f$ injective implies $f$ is monic. Injectivity is certainly not stronger than any other commonly used strengthening of the notion of monomorphism, because there are examples of injective maps that aren't extremal, and that's the weakest strengthening I know of. (An injective map between topological spaces is extremal monomorphism iff it is isomorphic to a subspace inclusion).
In the other direction, split monos are of course bound to be injective, because they are exactly the ones preserved by every functor, but it is easy to see that ie. regular monos don't need to be injective. Take the fork diagram, ie. a diagram $* → * ⇒ *$ such that the first map equalizes the other two, and it's obviously regular, but you can map it faithfully to something non-injective in $\mathrm{Set}$.
The situation with epimorphisms is similar.
If however the forgetful functor is representable (as is often the case, in particular when it's right adjoint) then it preserves limits, and therefore preserves all monos, meaning that all monos are injective (but epis needn't be surjective).
If the forgetful functor is monadic then it reflects regular epimorphisms (you can find the proof in Borceux's Handbook of Categorical Algebra), and since every surjection in $\mathrm{Set}$ is regular, this means that surjective morphism in a category which is monadic over $\mathrm{Set}$ is necessarily regular epi too (even though epis in general aren't, epimorphism $\mathbb Z → \mathbb Q$ of rings is a classical example).