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)?