[1]:http://pages.bangor.ac.uk/~mas010/pdffiles/brown-chnscoeff.pdf 

I gave a more elaborate example to the Universal Coefficient splitting being non natural in my paper  ``Cohomology with chains as coefficients'', _Proc. London Math. Soc_.  (3) 14 (1964), 545-565, available [here][1]. It is proved there  that for chain complexes $K,L$ which are free and are  zero  below dimension $0$,   there is an isomorphism for any abelian group $G$ 

$$H^*( K \otimes L, G) \cong H^*(K, H^*(L,G))$$ 

which can be chosen to be natural with respect to maps of $K$ but  **not** with regard to maps  of $L$,  nor in Example 3.2 maps of $G$. The naturality with respect to maps of $K$ is useful to recover R. Thom's  determination of the weak homotopy type of the function space $K(G,n)^Y$ and further  to determine $k^Y$ where $k$ is a cohomology operation.