---- > [!definition] Definition. ([[covariant functor]]) > Let $\mathsf{C}$, $\mathsf{D}$ be two [[category|categories]]. A **covariant functor** $\mathscr{F}:\mathsf{C}\to \mathsf{D}$ is an assignment of an object $\mathscr{F}(A) \in \text{Obj}(\mathsf{D})$ for every $A \in \text{Obj}(\mathsf{C})$ and of a function $\text{Hom}_{\mathsf{C}}(A,B) \to \text{Hom}_{\mathsf{D}}\big( \mathscr{F}(A), \mathscr{F}(B) \big)$ for every pair of objects $A,B$ in $\mathsf{C}$; said function is also denoted $\mathscr{F}$ must satisfy the so-called **functoriality properties** of respecting identities and compositions: > 1. $\mathscr{F}(1_{\mathsf{A}})=1_{\mathscr{F}(\mathsf{A})}$ for all $A \in \text{Obj}(\mathsf{A})$. > 2. $\mathscr{F}(\beta \circ \alpha)=\mathscr{F}(\beta) \circ \mathscr{F}(\alpha)$ for all $\alpha \in \text{Hom}_{\mathsf{C}}(A,B)$ and $\beta \in \text{Hom}_{\mathsf{C}}(B,C)$. > >Note that functors can be composed in the evident way. > **Notation.** Often we juxtapose rather than parenthesize: $\mathscr{F}A=\mathscr{F}(A)$ and $\mathscr{F}\alpha=\mathscr{F}(\alpha)$. ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```