Terminology on pullbacks

Question

I'm quite confused with the use of pullbacks, and in particular I wonder which terminology I shall use in the following examples. Let $X$ and $Y$ be arbitrary sets.

1. Suppose that $f,g:X\to Y$ and I know that there exists $h:X\to X$ such that $f = g\circ h$. Shall I say that $h$ is a pullback (of $f$? $g$?), or how shall I call $h$?

2. Suppose that $R$ is a relation on $Y$ and $f:X\to Y$ is some map. I define $S$ on $X$ by $$ xSx'\quad \iff \quad f(x)Rf(x'). $$ Shall I say that $S$ is a pullback of $R$ along $f$, or how shall I call $S$?

Answer 1

1. Just don't call it a pullback.

2. Yes this is ok.