Given commutative ring $k$, associative $k$-algebra $A$, and $A$-bimodule $M$, we can define the enveloping algebra $A^e:=A\otimes A^{op}$, where $A^{op}$ is the algebra $A$ but with multiplication reversed, i.e. $a\odot_{op}b:=ba$. Then any $A$-bimodule can be viewed as a left $A^e$-module, with operation $(a\otimes b)m:=amb$.
In the source below, we have isomorphism $\text{Hom}_{A^e}(A^{\otimes n},M)\cong\text{Hom}_k(A^{\otimes n-2},M),$ by sending $f\mapsto\tilde{f}$, where $\tilde{f}(x):=f(1\otimes x\otimes1).$ Given $\lambda\in k,$ $$\tilde{f}(\lambda x)=f(1\otimes\lambda x\otimes1)\neq\lambda f(1\otimes x\otimes 1)=\lambda\tilde{f}(x),$$ so in what sense is $\tilde{f}$ a $k$-homomorphism? And given $g\in\text{Hom}_k(A^{\otimes n-2},M),$ how do we define the isomorphism in reverse?
Source: Hochschild cohomology: some methods for computations, María Julia Redondo.