Let $A$ and $B$ be alegebras of any kind (the example that I have in mind is the underlying module of a graded operad) and consider its endomorphism operads $\mathrm{End}_A$ and $\mathrm{End}_B$, where $\mathrm{End}_X(n) = \hom(X^{\otimes n}, X)$. Given a map $f:A\to B$ I would like to be able to define a map $\mathrm{End}_A\to \mathrm{End}_B$ (or contravariantly if that was possible).
However, even for $n=1$ this only seems to be possible if $f$ is an isomorphism, since we would need to define a map $B\to B$ from a map $A\to A$, and this would generallyv be possible to achieve by means of $B\xrightarrow{f^{-1}} A\to A\xrightarrow{f}B$.
For general $n$ the situation is the same, as we would have to take an $n$-fold tensor product of $f^{-1}$.
Are there any more general conditions (either on the algebras or the map) for some kind of algebras where it is possible to stablish such maps even if $f$ is not an isomorphism (or more generally if $A$ and $B$ are not isomorphic)?
I know the question is very broad since I am not being very specific about the category in which I'm working, but I am really interested in any possibility and I may even have to define a category according to the suitable notion of morphism that allows the above maps.