Given a balanced bimodule $_SP_R$, is $R$ isomorphic to $\text{Hom}_S(P,P)$? For clarity's sake, let me recall some definitions:
Given two rings $R$ and $S$, we call a bimodule $_SP_R$ balanced if the ring homomorphisms
$$\lambda_P:S \rightarrow \text{End}(P_R)$$
$$\rho_P:R \rightarrow \text{End}(_SP)$$
defined via
$$\lambda_P(s)(x)=s\cdot x, \;\; \rho_P(r)(x)=x\cdot r$$
are surjective. We further call $_SP_R$ faithfully balanced if these two homomorphisms are isomorphisms.
In a proof I try to comprehend, at one point it seems that, assuming $_SP_R$ is balanced, the folowing isomorphism
$$R \simeq \text{Hom}_S(P,P)$$ should hold.
I can see the reason why it is so when I further assume that $_SP_R$ is faithfully balanced, but not in the more general case of "only" balanced bimodule.
Does this statement in general (i.e. for balanced bimodules) hold? If it does, how can it be proven?
Notes: 
1) The theorem I was referring to is Morita's characterization of equivalent rings. As it turns out, the balanced bimodule considered in the proof turns out to be faithfully balanced in the end. But the proof seems to use only the part that it is balanced - hence my confusion.
2) If I understand the rest of the proof correctly, the isomorphism has a meaning of "$R-R$-isomorphism", i.e. those objects should be isomorphic as left and right $R$-bimodules.
 A: The remark in your comment is correct: in general you need additional conditions on the balanced bimodule $P$ to establish the Morita equivalence you are referring to.
As an example of related discussion, let us
introduce the  Morita context (see Jacobson "Basic Algebra II", pag. 166) $(R,S,M,N,\tau,\mu)$ with $R$ and $S$ rings, $M$ $S$-$R$-bimodule, $N$ $R$-$S$-bimodule and  bimodule homomorphisms
$\tau: M\otimes_R N\rightarrow S$ and $\mu: N\otimes N_S M\rightarrow R$ satisfying a natural compatibility condition.  It can be shown (thm. "Morita I" loc. cit) that if $\tau$ and $\mu$ are surjective, then $\lambda: S\rightarrow End_R(M)$ and $\rho: R\rightarrow End_S(M)$ are ring isomorphisms ($\rho$ is a ring anti-isomorphism, to be precise) and $\tau$ and $\mu$ are actually bimodule isomorphisms.
The surjectivity of $\tau$ and $\mu$ is then important. An easy example is given by  the Morita context  $(R,S:=End_R(M),M,Hom_R(M,R),\tau,\mu)$ with $\tau: Hom_R(M,R)\otimes_S M\rightarrow R$ with $\tau(\phi,m):=\phi(m)$. Surjectivity (and even bijectivity) of $\tau$ is a non trivial condition that is satisfied, for example, when $M$  is a progenerator (loc. cit), as you suggested.
