If $X^\bullet$ and $Y^\bullet$ are chain complexes over a field, we know from Kunneth theorem that $$H^*((X\otimes Y)^\bullet)\cong H^*(X^\bullet)\otimes H^*(Y^\bullet) $$
I want to know if there is a similar theorem for internal homs: for chain complexes $X^\bullet$ and $Y^\bullet$, we set
$$[X,Y]^\bullet = \underset{k\in\mathbb Z}{\prod} [X^k,Y^{k+\bullet}]$$
Do we have ?
$$ H^*([X,Y]^\bullet)=[H^*(X^\bullet),H^*(Y^\bullet)]\qquad \mbox{or} \qquad H^n([X,Y])=\underset{j-i=n}{\prod}[H^i(X),H^j(Y)]$$
If this isn't true, is there some kind of relation between the two sides, in general or in some special cases such as with bounded complexes etc?
