---- > [!definition] Definition. ([[hom functor]]) > Let $\mathsf{C}$ be a [[category]], and let $X$ be an object of $\mathsf{C}$. Then the assignments $A \mapsto \text{Hom}_{\mathsf{C}}(X,A) \ \ , \ \ A \mapsto \text{Hom}_{\mathsf{C}}(A,X)$lead to, respectively, [[covariant functor|covariant]] and [[contravariant functor|contravariant]] functors $\mathsf{C} \to \mathsf{Set}$. These are denoted $\text{Hom}_{\mathsf{C}}(X, -)$ and $\text{Hom}_{\mathsf{C}}(-{, X})$ respectively. > In particular, $\text{Hom}_{\mathsf{C}}(X, -): \mathsf{ C} \to \mathsf{Set}$ maps each morphism $f:A \to B$ to the map $f \circ -$, i.e., $\begin{align} \text{Hom}_{\mathsf{C}}(X, A) & \to \text{Hom}_{\mathsf{C}}(X,B) \\ g & \mapsto f \circ g \end{align}.$ Meanwhile, $\text{Hom}_{\mathsf{C}}(-,X): \mathsf{C} \to \mathsf{Set}$ maps each morphism $h:A \to B$ to the map $\begin{align} \text{Hom}_{\mathsf{C}}(A, X) &\to \text{Hom}_{\mathsf{C}}(B,X) \\ g & \mapsto g \circ h \end{align}.$ > One can also consider the **hom [[bifunctor]]** $\text{Hom}_{\mathsf{C}}(-,-): \mathsf{C}^{\text{op}} \times \mathsf{C} \to \mathsf{Set}$ assigning the ordered pair $(C_{1}, C_{2})$ to $\text{Hom}_{\mathsf{C}}(C_{1}, C_{2})$ and the morphism $(f_{1}:C_{1}' \to C_{1}, f_{2}:C_{2} \to C_{2}')$ to $\begin{align} f_{2} \circ - \circ f_{1}: \text{Hom}_{\mathsf{C}}(C_{1}, C_{2}) &\to \text{Hom}_{\mathsf{C}}(C_{1}', C_{2}') \\ g& \mapsto (f_{2} \circ g \circ f_{1}) \end{align}$ ---- #### ---- #### 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 > ```