# Mappings that are not functors in category theory

Let C be a category and let $$F$$ be a mapping on C. Assume that $$F$$ is defined for all objects of C. However, on the arrows of C the mapping $$F$$ does not always behave like a (covariant) functor. For example, it may be the case that for some arrow $$f: x \rightarrow y$$ in C, the mapping $$F$$ returns an arrow $$F(f): F(x)\rightarrow F(y)$$ in C. For another arrow $$g:a \rightarrow b$$ in C, the mapping $$F$$ may return an arrow $$F(g): F(b)\rightarrow F(a)$$ in $${\bf C}^{op}$$. And possibly for other arrows of C, the mapping $$F$$ may not even be defined. Such a mapping is neither covariant not contravariant and it may also be partial. Is there a special name for such mappings?

• What's the example you have in mind? Dec 16, 2022 at 19:13
• Take for example a partially ordered set $(P,\sqsubseteq)$, which gives rise to a category. Consider a function $F: P \rightarrow P$. Such functions can be monotone, anti-monotone, or neither. It seems that in category theory we are mainly interested in covariant functors (ie., monotone functions) and contravariant functors (ie., anti-monotone ones). Is there a more general notion than just covariant/contravariant? Or is this just called a mapping''? I am new to category theory, so I apologize if my question sounds pedantic. Dec 17, 2022 at 14:50
• This simply corresponds to a function between the object-classes of the two categories. Dec 17, 2022 at 14:53
• But what if we are also interested in the effect that such a function has on arrows (and only on objects)? Dec 17, 2022 at 14:59
• But in these cases, the posetal structure is irrelevant. It's difficult to suggest a definition without any motivating examples. Dec 17, 2022 at 15:43