Relating categorical properties of arrows 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.


If it can help you, in The Joy of Cats pag 121 there is this nice diagram:

