The UC framework [Can00 (version of 2020-02-11)] defines security (defn 9) as for all adversaries there exists a simulator such that for all environments the environment output is indistinguishable in the ideal and real model.
$\forall A \exists S \forall E$:
$$EXEC_{\varphi,S,E} \approx EXEC_{\pi,A,E}$$
where $EXEC_{\pi,A,E} = \{EXEC_{\pi,A,E}(k,z)\}_{k \in \mathbb{N},z\in\{0,1\}^*}$. This means that a simulator must "fool" all environments on any input.
Claim 14 considers specialized simulators that can depend on the environment and states that the resulting definition for security is equivalent.
$\forall A \forall E \exists S$:
$$EXEC_{\phi,S,E} \approx EXEC_{\pi,A,E}$$
I am not following the proof.
assume that $π$ UC-emulates $φ$ with respect to specialized
simulators. That is, for any PPT adversary $A$ and PPT environment $E$ there exists a PPT simulator
$S$ such that $EXEC_{φ,S,E} ≈ EXEC_{π,A,E}$. Consider the “universal environment” $E_u$ which expects its
input to consist of $(\langle E \rangle, z, t)$, where $\langle E \rangle$ is an encoding of an ITM $E$, $z$ is an input to $E$, and $t$
is a bound on the running time of $E$. ($t$ is also the import of the input.) Then, $E_u$ runs $E$ on
input $z$ for up to $t$ steps, outputs whatever $E$ outputs, and halts. Clearly, machine $E_u$ is PPT.
(In fact, it runs in linear time in its input length.) We are thus guaranteed that there exists a
simulator $S$ such that $EXEC_{φ,S,E_u} ≈ EXEC_{π,A,E_u}$.
(emphasis mine)
I do not see why that last line holds.
Concretely, consider two environments $E'$ and $E''$, and let $S'$ be a specialized simulator for $E'$:
$$EXEC_{\varphi,S',E'} \approx EXEC_{\pi,A,E'}$$
but $S'$ is not a valid simulator for $E''$:
$$EXEC_{\varphi,S',E''} \not\approx EXEC_{\pi,A,E''}.$$
Then $S'$ "fools" $E_u$ on input $E'$:
$$EXEC_{\varphi,S',E_u}(k, (\langle E' \rangle, z, t)) \approx EXEC_{\pi,A,E_u}(k, (\langle E' \rangle, z, t))$$
but not on input $E''$:
$$EXEC_{\varphi,S',E_u}(k, (\langle E'' \rangle, z, t)) \not\approx EXEC_{\pi,A,E_u}(k, (\langle E'' \rangle, z, t))$$
and thus
$$EXEC_{\varphi,S',E_u} \not\approx EXEC_{\pi,A,E_u}$$
because it has to fool the environment on all inputs. Did I find a mistake or am I misunderstanding something?
