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The induced internal hom for modules over a Hopf algebra on ${\mathrm{Hom}}_k(H,H)$ iswhere both occurrences of $H$ have their left regular action. If $T$ is a morphism of left $H$-modules, thenSo every such morphism is an invariant vector of a module over a Hopf algebra.

For the converse, a calculation valid for every $k$-linear $T$ isIf $T$ is invariant, its left-hand side is instead $\sum \epsilon (h_{(1)})T(h_{(2)}x)=T(hx)$. Hence $T(hx)=hT(x)$ for all $h,x$, which is precisely the R-module homomorphism condition.

In fact an endomorphism of the left regular module is determined by $a=T(1)$ and has $T(x)=xa$. Conversely each right multiplication map is left $H$-linear. Thus the invariant space has the explicit descriptionAs a vector space it is isomorphic to $H$, and under composition its algebra is the opposite algebra $H^{\mathrm{op}}$. No dimension or semisimplicity hypothesis is needed.