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?
One has to read $$C \xrightarrow {x} X {\rm \ for \ any \ object\ } C\in \mathcal{C}$$ instead of $$\mathcal{C}\xrightarrow {x} X.$$
A proof that "strict epimorphism + monomorphism = isomorphism" is in the book
I. Bucur, A. Deleanu, Introduction to the Theory of Categories and Functors, 1970, Lemma 3.17.