# Show that is a covariant functor

Let $C'$ be any category and $C$ be the category of sets. Let $A$ be a fixed element of $C'$. Show that $X\mapsto Hom(X,A)$ define a covariant functor $F:C'\rightarrow C$.

My problem is, when I have $f:P\rightarrow Q$ be a morfismo in $C'$, if $h:P\rightarrow A$, how can I define $F(f):Q\rightarrow A$?

Thanks

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It is the other way. For a morphism $h:Q\to A$ you want to assign a morphism $F(f)(h):P\to A$. The functor $X\mapsto Hom(X,A)$ is contravariant. – Stefan Hamcke Jan 10 '14 at 1:54

Given a morphism $f: P \longrightarrow Q$ in $C'$, you want to define a morphism $$F(f): \mathrm{Hom}(Q, A) \longrightarrow \mathrm{Hom}(P, A)$$ in $C$. This means that given a morphism $h: Q \longrightarrow A$, we should get another morphism $F(f)(h): P \longrightarrow A$. Just define $$F(f)(h) = h \circ f.$$ You can easily verify that $$F(g \circ f) = F(f) \circ F(g)$$ and $$F(\mathrm{Id}_P) = \mathrm{Id}_{\mathrm{Hom}(P,A)},$$ so that $F$ is indeed a contravariant functor.