> [!definition] > > Let $\mathfrak{U}, \mathfrak{U}'$ be [[Category|categories]]. A **covariant functor** $\lambda: \mathfrak{U} \to \mathfrak{U}'$ is a map that associates > - Each object $X \in \text{Ob}(\mathfrak{U})$ to an object $\lambda(X) \in \text{Ob}(\mathfrak{U})$, > - Each morphism $f: X \to Y$ to a morphism $\lambda(f): \lambda(X) \to \lambda(Y)$. > > Such that > > - Whenever $f, g$ are morphisms in $\mathfrak{U}$ that can be composed $\lambda(fg) = \lambda(f)\lambda(g)$. > - $\lambda(\text{id}_X) = \text{id}_{\lambda(X)}$ for all $X \in \text{Ob}(\mathfrak{U})$. > [!definition] > > A functor $\lambda$ is *covariant* if $\lambda(fg) = \lambda(f)\lambda(g)$ whenever they can be composed, and *contravariant* if $\lambda(fg) = \lambda(g)\lambda(f)$. > [!definition] > > Let $\mathfrak{U}, \mathfrak{U}'$ be categories. The functors from $\mathfrak{U}$ to $\mathfrak{U}'$ of the same variance form a category $\text{Fun}(\mathfrak{U}, \mathfrak{U}')$, whose morphisms are called the *natural transformations*. > > More specifically, let $\lambda, \mu$ be two covariant functors from $\mathfrak{U}$ to $\mathfrak{U}'$, then a natural transformation $t: \lambda \to \mu$ consists of a collection of morphisms > $ > t_X: \lambda(X) \to \mu(X) > $ > with $X$ ranging over $\text{Ob}(\mathfrak U)$ such that the following diagram commutes > $ > \begin{CD} > \lambda(X) @>{t_X}>> \mu(X)\\ > @V{\lambda(f)}VV @VV{\mu(f)}V \\ > \lambda(Y) @>>{t_Y}> \mu(Y) > \end{CD} > $ > for any $f: X \to Y$ in $\mathfrak U$.