The Universal Coefficient Theorem for, say, singular cohomology should give examples. For any abelian group $G$ and $n> 0$, the functors from spaces to abelian groups given by $$X\mapsto H^n(X;G),\qquad X\mapsto \mathrm{Ext}(H_{n-1}(X),G)\oplus\mathrm{Hom}(H_n(X),G)$$ are isomorphic, but not naturally so. See Hatcher's "Algebraic Topology", Chapter 3.1 (in particular Exercise 11 at the end of that section).