I want to write some short introduction of Morita theory, since it seems that not all books introduce this classic thing in the following easy and brief way.
The quotient relation.
We know the adjointness
$$Hom_B^C(Y,Hom_A(X,Z))=Hom_A^C(Xotimes_BY,Z)=Hom_A^B(X,Hom^C(Y,Z)). $$
There is other isomorphisms
$$egin{array}{c}Hom_A(X,Y_B)otimes_B Z o Hom_A(X,Yotimes_B Z),\Xotimes_A Hom_B(Y_A,Z) o Hom_B(Hom^A(X,Y_A),Z). end{array}$$
For the first map. When $X=A$, it is isomorphic trivially. When $X=A^n$, it is isomorphic trivially. When $X$ is f.g. projective, then as a summand of some $A^n$ it is isomorphic. When $X$ is f.pre. and $Z$ is flat, then it is isomorphic by five lemma.
For the second map. When $X=A$, it is isomorphic trivially. When $X=A^n$, it is isomorphic trivially. When $X$ is f.g. projective, then as a summand of some $A^n$ it is isomorphic.When $X$ is f.pre. and $Z$ is injective, then it is isomorphic by five lemma.
And the dual for right modules
$$egin{array}{c}Xotimes_A Hom^B(Y,{}_{A}Z) o Hom^B(X,Zotimes_AY),\Hom^A(X_B,Y)otimes_B Z o Hom^A(Hom_B(X,Z),Y).end{array}$$
Left and right adjoint functor of tensor.
For a $Pin B extrm{-}mathsf{mod} extrm{-} A$, and projective as $B$-module, we can consider $Q=Hom_B(P,B)$. Now
$$Qotimes_B - = Hom_B(P,B)otimes_B -=Hom_B(P,-)$$
is an adjoint of $Potimes_A - $.
We can also consider $Q'=Hom^A(P,A)$, now
$$egin{array}{rl}Hom_A(Q'otimes_B U, V) & = Hom_B(U,Hom_A(Q', V))\& = Hom_B(U,Hom_A(Hom^A(P,A),V))\& =Hom_B(U,Potimes_A Hom_A(A,V))\& = Hom_B(U,Potimes_A V). end{array}$$
But we don't use it.
Equivalence $Rightarrow$ Projective generator.
Assume $A extrm{-}mathsf{mod} cong B extrm{-}mathsf{mod}$. Consider the image of $A$ in $B extrm{-}mathsf{mod}$, say $P$. Then $P$ is a projective generator (any $Min B extrm{-}mathsf{mod}$ is a quotient of some $P^n$). Furthermore, $P$ is also f.g. since $Hom_B(P,-)$ commutes with direct sum. Now, $Hom_B(P,P)=Hom_A(A,A)=A^{op}$, so $Pin B extrm{-}mathsf{mod} extrm{-} A$. Then the functor $A extrm{-}mathsf{mod} o B extrm{-}mathsf{mod}$ is given by $Potimes_A-$.
Since the quasi-equivalence is adjoint, and adjoint is unique, so $Qotimes-$ is the desired inverse, where$Q=Hom_B(P,B)$ or $Q=Hom^A(P,A)$ works (but they are isomorphic by uniqueness).
Projective generators $Rightarrow$ Equivalence.
Next, if we have a projective generator $P$ in $B extrm{-}mathsf{mod}$, denote $A=End_B(P,P)^{op}$, and consider $Q=Hom_B(P,B)$, then we have two maps
$$Qotimes_B P=Hom_B(P,B)otimes_B P=Hom_B(P,P)= Aqquad : eta$$
and
$$epsilon: qquad Potimes_A Q=Potimes_B Hom_B(P,B) o B. $$
Note that
$$P=Potimes Acong Potimes_A Qotimes_B P o Botimes P=P$$
is identity, so $epsilonotimes P$ is an isomorphism, but because $P$ is projective generator, so $Potimes_AQotimes_B B o B$ is isomorphism. So $Potimes-$ and $Qotimes-$ induces equivalence.
Morita context.
We will say a four tuple $(A,B,P,Q)$ with $A,B$ algebras and $P,Q$ bimodules is a Morita context if
$$epsilon: Potimes_A Q o Bqquad eta:A o Qotimes_B P$$
are isomorphisms as bimodule, and satisfy the
$$(Potimes eta)circ (epsilonotimes P)=operatorname{id}_p,qquad (etaotimes Q)circ (Qotimes epsilon)=operatorname{id}_Q. $$
Of course, this defines an adjoint. So our discussion above shows that the equivalences between $A extrm{-}mathsf{mod}$ and $B extrm{-}mathsf{mod}$ are all from Morita type.
Note that Morita context is symmetric, not only left modules but also right modules (even the bimodules)!As a result, $A extrm{-}mathsf{mod}cong B extrm{-}mathsf{mod} iff mathsf{mod} extrm{-} Acong mathsf{mod} extrm{-} B$.
Then $P$ are generators of both sides. Because we can take some surjective $A^n ightarrow Q$, then $P^n=Potimes_AA^n ightarrow Potimes Qcong B$. So
$$egin{array}{rl}Q & = Hom_A(A,Q) = Hom_A(Qotimes_B P,Q)=Hom_B(P,Potimes_A Q)\& =Hom_B(P,B). end{array}$$
The same reason,
$$egin{array}{rl}Q & = Hom^B(B,Q) = Hom^B(Potimes_A Q,Q)=Hom^A(Q,Qotimes_B P)\& = Hom^A(Q,A).end{array}$$
And as algebras
$$egin{array}{rl}End_B(P) & =Hom_B(P,P)=Hom_A(A,Qotimes P) =A^{op}\End^A(P) & =Hom^A(P,P)=Hom^B(B,Potimes Q) =B end{array}$$