Specialized simulators in Universal composability

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?

The point that you are missing is as follows. If a protocol is UC secure for specialized simulators, then $$\forall A \forall E\exists S$$. In particular, this is true for the universal environment $$E_u$$. Denote this simulator by $$S_u$$. The argument is that $$S_u$$ is a simulator for all environments. In particular, $$S_u$$ working with $$E_u$$ on input $$(\langle E\rangle,z,t)$$ is exactly the same as $$S_u$$ working directly with the environment $$E$$.