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What does Variance of Functor mean? I have to show that composition of two functor is covariant if the two functor has same variance, and composition of functors is contravariant if they have different variance.

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2 Answers 2

Functors map both objects and morphisms. Covariant functors preserve morphism composition: $F(f \circ g) = F(f) \circ F(g)$. However, contravariant functors reverse this composition: $F(f \circ g) = F(g) \circ F(f)$.

So if you compose two functors of the same variance, you'll either get preserve + preserve = preserve or flip + flip = preserve (flip twice and you're back in order).

On the other hand if two functors have opposite variance, you'll have preserve + flip = flip (flipping once and preserve order once will result in flipping the order around).

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+1 very well put :-) –  magma Mar 7 at 9:42

"Variance" means the same type: either covariant ar contravariant.

So you are asked to show that the composition of two covariant or two contravariant functors is covarient, and that composition of a covariant functor with contravariant (in any order) is contravariant.

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