Illustration of Hom(A,−){\displaystyle Hom(A,-)} functor: Fix A∈C{\displaystyle A\in {\mathcal {C}}}. Then there are sets of morphisms Hom(A,X)={α1,α2,⋯}{\displaystyle Hom(A,X)=\{\alpha _{1},\alpha _{2},\cdots \}}, Hom(A,Y)={β1,β2,⋯}{\displaystyle Hom(A,Y)=\{\beta _{1},\beta _{2},\cdots \}}, etc. Also, for f:X→Y{\displaystyle f:X\rightarrow Y}, we define a map Hom(A,f):Hom(A,X)→Hom(A,Y){\displaystyle Hom(A,f):Hom(A,X)\rightarrow Hom(A,Y)}, i.e. Hom(A,f):{α1,α2,⋯}→{β1,β2,⋯}{\displaystyle Hom(A,f):\{\alpha _{1},\alpha _{2},\cdots \}\rightarrow \{\beta _{1},\beta _{2},\cdots \}} by α↦f∘α{\displaystyle \alpha \mapsto f\circ \alpha } (illustrated by the dashed arrow). Thus Hom(A,−){\displaystyle Hom(A,-)} is a covariant functor.
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