Hom funktor je kovariantní bifunktor v lokálně malé kategorii C{\displaystyle {\mathcal {C}}} typu Cop×C→Set{\displaystyle {\mathcal {C}}^{op}\times {\mathcal {C}}\rightarrow \mathbf {Set} } definovaný pro A,B∈Ob(C){\displaystyle A,B\in \mathrm {Ob} ({\mathcal {C}})} takto:
Hom(A,−){\displaystyle Hom(A,-)} je kovariantní a pro X,Y∈Ob(C),f:X→Y{\displaystyle X,Y\in \mathrm {Ob} ({\mathcal {C}}),f:X\rightarrow Y} je Hom(A,f){\displaystyle Hom(A,f)} funkce Hom(A,f):Hom(A,X)→Hom(A,Y),Hom(A,f)(g)=f∘g{\displaystyle Hom(A,f):Hom(A,X)\rightarrow Hom(A,Y),Hom(A,f)(g)=f\circ g}, kde g∈Hom(A,X){\displaystyle g\in Hom(A,X)}.
Podobně Hom(−,B){\displaystyle Hom(-,B)} je kontravariantní a pro X,Y∈Ob(C),h:X→Y{\displaystyle X,Y\in \mathrm {Ob} ({\mathcal {C}}),h:X\rightarrow Y} je Hom(h,B){\displaystyle Hom(h,B)} funkce Hom(h,B):Hom(Y,B)→Hom(X,B),Hom(h,B)(g)=g∘h{\displaystyle Hom(h,B):Hom(Y,B)\rightarrow Hom(X,B),Hom(h,B)(g)=g\circ h}, kde g∈Hom(Y,B){\displaystyle g\in Hom(Y,B)}.
Hom(−,−){\displaystyle Hom(-,-)} je tedy kovariantní bifunktor Cop×C→Set{\displaystyle {\mathcal {C}}^{\mathrm {op} }\times {\mathcal {C}}\rightarrow \mathbf {Set} }.
