I was wondering if the following two meanings of pullback are related and how:
a function $f$ of a variable $y$, where $y$ itself is a function of another variable $x$, may be written as a function of $x$. Then $f(y(x)) \equiv g(x)$ is the pullback of $f$ by the function $y(x)$.
the pullback of the morphisms $f$ and $g$ consists of an object $P$ and two morphisms $p_1 : P \rightarrow X$ and $p_2 : P \rightarrow Y$ for which the diagram
commutes. Moreover, the pullback $(P, p_1, p_2)$ must be universal with respect to this diagram.
- Also, is it possible to define pushforward/pushout in terms of composition of functions?
Thanks and regards!