I have got a very basic question, but it would simplify some things, so I hope this resolves in either the affirmative or maybe someone can provide an I guess simple non-example.
Given pairs of Hilbert C*-modules $X_1$ and $X_2$ over a common C*-algebra as well as $Y_1$ and $Y_2$ over a possibly different C*-algebra. Suppose the adjointable operators of those pairs are *-isomorphic, $$ \Phi:\mathcal{L}(X_1)\cong\mathcal{L}(X_2),\qquad\Psi:\mathcal{L}(Y_1)\cong\mathcal{L}(Y_2). $$
My first question: Are the adjointable operators in between *-isomorphic, $$ \mathcal{L}(X_1,Y_1)\cong\mathcal{L}(X_2,Y_2)? $$
My second question: Are the inner *-homomorphisms $$ \operatorname{ad}(R_1):\mathcal{L}(X_1)\to\mathcal{L}(Y_1):\quad\operatorname{ad}(R_1)(T_1):=R_1T_1R_1^* $$ for isometries $R_1:X_1\to Y_1$, in one-to-one correspondence via $$ \operatorname{ad}(R_1)\mapsto\Psi\circ\operatorname{ad}(R_1)\circ\Phi^{-1}? $$