Is this notation on the restriction of a function in group theory common?

Question

If $f: X \rightarrow Y$ is a function between sets $X$ and $Y$, then a common notation to use when we want to restrict $f$ to a certain domain $X' \subset X$ is $f|_{X'}: X' \rightarrow Y$.

I'm doing some group theory and have come across the following:

Let $G,K$ be groups, $\phi: G \rightarrow K$ a homomorphism, $H \leq G$ a subgroup. Consider $\phi_H$...

Is this common notation (perhaps in group theory)? It wasn't immediately clear to me that this was a restriction to $H$, i.e. $\phi_H: H \rightarrow K$ until I had read the paragraph to the end.

Answer 1

This is not a standard notation, though I have seen notation like this used occasionally (not specifically in group theory) when you need to repeatedly refer to lots of restrictions in the course of a single argument. I would never use a notation like this without defining it for my readers. The standard notation is $\phi|_H$, not $\phi_H$.

Answer 2

I think the restriction $\phi_H$ to $H$ indicated is not only restriction of application, but it seems more, this is a group homomorphism