TITLE: Pushforward and pullback.
QUESTION [20 upvotes]: Often I say "pushforward" or "pullback", but I do not know exact
meaning of these words.
Each time I see a map $f\colon X \to Y$, plus:

I have some object $O_X$ associated with $X$ (say measure or subset);
the map $f$ gives me a natural way to find
corresponding object $O_Y$ associated with $Y$.

Then I say "$O_Y$ is pushforward of $O_X$" and I write $O_Y=f_*O_X$.
If I switch $X$ and $Y$ in (1) and (2), I say "$O_X$ is pullback of
$O_Y$" and I write $O_X=f^*O_Y$.
I do this all the time, and no one complains, but I do not feel that
it is right...

Could someone explain the right way to think about "pushforward" and "pullback"?

REPLY [4 votes]: I think that in general, the word "pullback" is associated to (some) contravariant functors, and "pushforward" is associated to (some) covariant functors. 
To be more specific, consider the measure example. There is an endfunctor $M$ of the category $Set$, which maps a set $X$ to the set $M(X)$ of all measures on $X$, and a function $f:X\to Y$ to a function $M(f)=f_*:M(X)\to M(Y)$, defined so that $f_*(\mu)$ is a measure on $Y$, given by $f_*(\mu)(U)=\mu(f^{-1}(U))$ for all $U\subseteq Y$ for which the right-hand side is defined. Since $M$ is a covariant functor, we might call $M(f)(\mu)=f_*(\mu)$ a pushforward of a measure $\mu$ on $X$. 
As an example of pullback, consider the pullback from cohomology. 
There is the functor $H^*$ from the category $Top$ of topological spaces to $Set$, which maps a topological space $X$ to the set (in fact, a graded ring) $H^*(X)$ of cohomology classes of $X$, and a continuous function $f:X\to Y$ to the function (in fact, a homomorphism of graded rings) $H^*(f)=f^*:H^*(Y)\to H^*(X)$ on cohomology classes. Since $H^*$ is a contravariant functor, we may call $H^*(f)(\sigma)=f^*(\sigma)$ a pullback of a cohomology class $\sigma$ on $Y$.