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