# Is it wrong to say “pushforward” and “pullback” for general functors?

Is it wrong ot use the terms pushforward (resp. pullback) about the induced map of general covariant (resp. contravariant) functors and denote them $\varphi_*$ (resp. $\varphi^*$)?

The terminology is standard in the case of e.g. sheaves, but over time I have found myself using them in this more general setting, and I would like to know if it should be considered wrong (or at least very far from standard)?