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. 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.

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. 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 (see the paper ``On Kunneth suspensions'', Proc. Camb. Phil. Soc. 60 (1964) 713-720, available here.

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. 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 also to determine $k^Y$ where $k$ is a cohomology operation.

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. 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.