I am trying to understand the meaning of a pushforward in the simplest context possible where the functions involved are defined on sets. From my readings in differential geometry, I have arrived at the following understanding that I have attempted to codify in a precise definition. Unfortunately, I have not been able to find a reference that defines the pushforward in this minimal context. My proposed definition is as follows:
Let $\phi:X \rightarrow Y$ be a bijection and let $f:X\rightarrow Z$ be any function from $X$ to the set $Z$. Then, the pushforward of $f$ by $\phi$ is a map $ \phi_*:Z^X \rightarrow Z^Y $ defined by $ \phi_* f := f \circ \phi^{-1}. $
So my question: Is this definition correct and is this is the right way to think of pushforward when only maps between sets are involved?