# Functor

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A functor is a type of map between two categories.

More precisely, a functor $F:\mathcal{C} \to \mathcal{D}$ is a mapping which

• sends every object $X$ of $\mathcal{C}$ to and object $F(X)$ of $\mathcal{D}$.
• sends every morphism $f:X\to Y$ of $\mathcal{C}$ to a morphism $F(f):F(X)\to F(Y)$ of $\mathcal{D}$.

Which satisfies the conditions:

• $F(1_X) = 1_{F(X)}$ for all $X\in \text{Ob}(\mathcal{C})$.
• $F(g\circ f) = F(g)\circ F(f)$ for all morphisms $f:X\to Y$ and $g:Y \to Z$ of $\mathcal{C}$.