Hamana's proof (Theorem 3.5 in Injective Envelopes of Operator Systems, PubL RIMS, Kyoto Univ. 15 (1979), 773-785) is fairly direct. Consider the partial ordering on the space $\Xi = \{ \phi \in UCP(W, W) \mid \phi \kappa = \kappa \}$ given by $\phi \prec \psi$ when $\| \phi(x) \| \leq \| \psi(x) \|$ for all $x \in W$. First, note that every element in $\Xi$ dominates a minimal element in $\Xi$ since if $\{ \phi_i \}_i$ is a decreasing net in $\Xi$, then taking any concrete realization $W \subset \mathcal B(\mathcal H)$ we have $E \phi \prec \phi_i$ for all $i$, where $E: \mathcal B(\mathcal H) \to W$ is any ucp idempotent and $\phi$ is any limit point of $\{ \phi_i \}_i$ in $UCP(W, \mathcal B(\mathcal H))$, which is compact in the topology of pointwise ultraweak convergence.

Second, note that any minimal element $\phi \in \Xi$ is an idempotent. Indeed, for any $N \geq 1$ and $x \in W$ we have $\| \frac{1}{N} \sum_{n = 1}^N \phi^n( x ) \| \leq \| \phi(x) \|$ and so by minimality it follows that this is equality. Considering $x = y - \phi(y)$ we have $\| \phi(y) - \phi^2(y) \| = \lim_{N \to \infty} \| \frac{1}{N} \sum_{n = 1}^N \phi^n(y - \phi(y) ) \| \leq \frac{2}{N} \| y \| \to 0$.

Thus, if every idempotent in $\Xi$ is the identity, it follows that the identity is the only map in $\Xi$.

Hamana's proof (Theorem 3.5 in Injective Envelopes of Operator Systems, PubL RIMS, Kyoto Univ. 15 (1979), 773-785) is fairly direct. Consider the partial ordering on the space $\Xi = \{ \phi \in UCP(W, W) \mid \phi \kappa = \kappa \}$ given by $\phi \prec \psi$ when $\| \phi(x) \| \leq \| \psi(x) \|$ for all $x \in W$. First, note that every element in $\Xi$ dominates a minimal element in $\Xi$ since if $\{ \phi_i \}_i$ is a decreasing net in $\Xi$, then taking any concrete realization $W \subset \mathcal B(\mathcal H)$ we have $E \phi \prec \phi_i$ for all $i$, where $E: \mathcal B(\mathcal H) \to W$ is any ucp idempotent and $\phi$ is any limit point of $\{ \phi_i \}_i$ in $UCP(W, \mathcal B(\mathcal H))$, which is compact in the topology of pointwise ultraweak convergence.

Second, note that any minimal element $\phi \in \Xi$ is an idempotent. Indeed, for any $N \geq 1$ and $x \in W$ we have $\| \frac{1}{N} \sum_{n = 1}^N \phi^n( x ) \| \leq \| \phi(x) \|$ and so by minimality it follows that this is equality. Considering $x = y - \phi(y)$ we have $\| \phi(y) - \phi^2(y) \| = \| \frac{1}{N} \sum_{n = 1}^N \phi^n(y - \phi(y) ) \| \leq \frac{2}{N} \| y \| \to 0$.

Thus, if every idempotent in $\Xi$ is the identity, it follows that the identity is the only map in $\Xi$.