Functor

Remarks

Such a functor is also called covariant. Functors compose, and every category has an identity functor. A functor sends isomorphisms to isomorphisms: if $g \circ f = 1_A$ and $f \circ g = 1_B$, then $F(g) \circ F(f) = 1_{F(A)}$ and $F(f) \circ F(g) = 1_{F(B)}$. Example: the forgetful functor from groups to sets sends a group to its underlying set and a group homomorphism to the same map. Both conditions hold because identities and composites of homomorphisms are the identity and composite maps. Non-example: sending each group $G$ to its centre $Z(G)$ and each homomorphism to its restriction does not define a functor from groups to groups. Let $f \colon \Z/2 \to S_3$ be the inclusion onto the subgroup generated by a transposition. It maps the centre $\Z/2$ outside the trivial centre of $S_3$. So restriction gives no morphism $Z(\Z/2) \to Z(S_3)$, and the second item of the data fails.