$\require{AMScd}$ I have the following question:
Let $\mathscr{C}$ be a category, $X,Y,Z\in Ob(\mathscr{C}), \ f\in Mor(X,Y),\ g\in Mor(Y,Z)$ and $h\in Mor(X,Z)$.
Question: What does the equality sign in the following diagram mean?
$$ \begin{CD} X @>f>> Y\\ @| @VVgV \\ X @>>h> Z \end{CD} $$
Does the equality sign only emphasize that the two objects are indeed the same, or does it mean that we have $X\stackrel{id}{\rightarrow}X$ or does it mean that we have an arbitrary isomorphism $X\stackrel{\sim}{\rightarrow}X$?
Thank you very much!
It represents the identity map. The notation also implies that you could think of the diagram as being the same as a triangle, with the two $X$s being squished together (kind of like a quotient of the diagram or something).