# Definition of strict epimorphism

Following is the definition of strict epimorphism from the paper I'm reading.

However, I have some confusion. What does $$\mathcal{C}\xrightarrow{x}X$$

mean? I think $\mathcal{C}$ is a category which contains $X$ as an object, so what is $x$?

And after this definition the author says that strict epimorphism + monomorphism = isomorphism. Could anyone provide me a proof?

I'm new to category theory, forgive me if the question is stupid.

By the way, I have googled the term strict epimorphism, seeing that there is few result. Even Wiki does not have this term. Is it an isolated term?

-
Shouldn't it be $g$ compatible if for all $x,y$, if $fx=fy$ then $gx=gy$? –  Stefan Hamcke Oct 30 '13 at 16:27
Otherwise any monic $g$ would be compatible. –  Stefan Hamcke Oct 30 '13 at 16:33
That would also be consistent with the nLab-definition. –  Stefan Hamcke Oct 30 '13 at 16:39

One has to read $$C \xrightarrow {x} X {\rm \ for \ any \ object\ } C\in \mathcal{C}$$ instead of $$\mathcal{C}\xrightarrow {x} X.$$