The point is that the $\omega$ in an $\omega$-model satisfies full induction, even if the whole model does not.

Take a nonstandard $\omega$-model $M\models\rm \Pi_n\text{-}collection+V\,{=}\,L$. Let $I$ be that given by Corollary 4.5. Suppose $\varphi(v,x)$ is a $\Pi_n$-formula for which $$A=\{x\in\mathrm L_I^M:\mathrm L_I^M\models\exists v\ \varphi(v,x)\}$$ is nonempty but has no $\in^M$-minimal element. By recursion, we will define an $\in^M$-decreasing $\omega$-sequence $(a_i)_{i\in\omega}$ of elements of $A$ such that $\{a_i:i\in\omega\}$ is $\Delta_{n+1}$-definable in $M$. This gives what we want because it contradicts $\Delta_{n+1}$-foundation and hence $\Pi_n$-collection in $M$.

Start with any $a_0\in A$. Suppose we already have $a_i\in A$, where $i\in\omega$. By our hypothesis, we know $A$ contains some $\hat a_{i+1}\in^M a_i$. Find $\hat v_{i+1}\in\mathrm L_I^M$ such that $M\models\varphi(\hat v_{i+1},\hat a_{i+1})$. Given any big enough $\mathrm L_\alpha^M\subseteq\mathrm L_I^M$ containing both $\hat a_{i+1}$ and $\hat v_{i+1}$, define $$ \begin{aligned} v_{i+1}&=\min\{ v\in\mathrm L_\alpha^M: M\models\exists x\in a_i\cap\mathrm L_\alpha^M\ \varphi(v,x) \},\ \text{and}\\ a_{i+1}&=\min\{ x\in\mathrm L_\alpha^M:M\models x\in a_i\wedge\varphi(v_{i+1},x) \}, \end{aligned} $$ where the minima are taken with respect to the $\mathrm L^M$-order. These exist because $M$ has $\Delta_{n+1}$-foundation. It can be verified that this choice of $a_{i+1}$ does not depend on the choice of $\alpha$.

Since $\omega^M=\omega$, one sees that $i\mapsto a_i$ is a total function $\omega^M\to M$. The rest is straightforward.

The point is that the $\omega$ in an $\omega$-model satisfies full induction, even if the whole model does not.

Take a nonstandard $\omega$-model $M\models\rm \Pi_n\text{-}collection+V\,{=}\,L$. Let $I$ be that given by Corollary 4.5. Suppose $\varphi(v,x)$ is a $\Pi_n$-formula for which $$A=\{x\in\mathrm L_I^M:\mathrm L_I^M\models\exists v\ \varphi(v,x)\}$$ is nonempty but has no $\in^M$-minimal element. By recursion, we will define an $\in^M$-decreasing $\omega$-sequence $(a_i)_{i\in\omega}$ of elements of $A$ such that $\{a_i:i\in\omega\}$ is $\Delta_{n+1}$-definable in $M$. This gives what we want because it contradicts $\Delta_{n+1}$-foundation and hence $\Pi_n$-collection in $M$.

Start with any $a_0\in A$. Suppose we already have $a_i\in A$, where $i\in\omega$. By our hypothesis, we know $A$ contains some $\hat a_{i+1}\in^M a_i$. Find $\hat v_{i+1}\in\mathrm L_I^M$ such that $M\models\varphi(\hat v_{i+1},\hat a_{i+1})$. Given any big enough $\mathrm L_\alpha^M\subseteq\mathrm L_I^M$ containing both $a_i$ and $\hat v_{i+1}$, define $$ \begin{aligned} v_{i+1}&=\min\{ v\in\mathrm L_\alpha^M: M\models\exists x\in a_i\ \varphi(v,x) \},\ \text{and}\\ a_{i+1}&=\min\{ x\in a_i:M\models\varphi(v_{i+1},x) \}, \end{aligned} $$ where the minima are taken with respect to the $\mathrm L^M$-order. These exist because $M$ has $\Delta_{n+1}$-foundation. It can be verified that this choice of $a_{i+1}$ does not depend on the choice of $\alpha$.

Since $\omega^M=\omega$, one sees that $i\mapsto a_i$ is a total function $\omega^M\to M$. The rest is straightforward.

The point is that the $\omega$ in an $\omega$-model satisfies full induction, even if the whole model does not.

Take a nonstandard $\omega$-model $M\models\rm \Pi_n\text{-}collection+V\,{=}\,L$. Let $I$ be that given by Corollary 4.5. Suppose $\varphi(v,x)$ is a $\Pi_n$-formula for which $$A=\{x\in\mathrm L_I^M:\mathrm L_I^M\models\exists v\ \varphi(v,x)\}$$ is nonempty but has no $\in^M$-minimal element. By recursion, we will define an $\in^M$-decreasing $\omega$-sequence $(a_i)_{i\in\omega}$ of elements of $A$ such that $\{a_i:i\in\omega\}$ is $\Delta_{n+1}$-definable in $M$. This gives what we want because it contradicts $\Delta_{n+1}$-foundation and hence $\Pi_n$-collection in $M$.

Start with any $a_0\in A$. Suppose we already have $a_i\in A$, where $i\in\omega$. By our hypothesis, we know $A$ contains some $\hat a_{i+1}\in^M a_i$. Find $\hat v_{i+1}\in\mathrm L_{\alpha_i}^M$ such that $M\models\varphi(\hat v_{i+1},\hat a_{i+1})$. Given any big enough $\mathrm L_\alpha^M\subseteq\mathrm L_I^M$ containing both $\hat a_{i+1}$ and $\hat v_{i+1}$, define $$ \begin{aligned} v_{i+1}&=\min\{ v\in\mathrm L_\alpha^M: M\models\exists x\in a_i\cap\mathrm L_\alpha^M\ \varphi(v,x) \},\ \text{and}\\ a_{i+1}&=\min\{ x\in\mathrm L_\alpha^M:M\models x\in a_i\wedge\varphi(v_{i+1},x) \}, \end{aligned} $$ where the minima are taken with respect to the $\mathrm L^M$-order. These exist because $M$ has $\Delta_{n+1}$-foundation. It can be verified that this choice of $a_{i+1}$ does not depend on the choice of $\alpha$.

Since $\omega^M=\omega$, one sees that $i\mapsto a_i$ is a total function $\omega^M\to M$. The rest is straightforward.

The point is that the $\omega$ in an $\omega$-model satisfies full induction, even if the whole model does not.

Take a nonstandard $\omega$-model $M\models\rm \Pi_n\text{-}collection+V\,{=}\,L$. Let $I$ be that given by Corollary 4.5. Suppose $\varphi(v,x)$ is a $\Pi_n$-formula for which $$A=\{x\in\mathrm L_I^M:\mathrm L_I^M\models\exists v\ \varphi(v,x)\}$$ is nonempty but has no $\in^M$-minimal element. By recursion, we will define an $\in^M$-decreasing $\omega$-sequence $(a_i)_{i\in\omega}$ of elements of $A$ such that $\{a_i:i\in\omega\}$ is $\Delta_{n+1}$-definable in $M$. This gives what we want because it contradicts $\Delta_{n+1}$-foundation and hence $\Pi_n$-collection in $M$.

Start with any $a_0\in A$. Suppose we already have $a_i\in A$, where $i\in\omega$. By our hypothesis, we know $A$ contains some $\hat a_{i+1}\in^M a_i$. Find $\hat v_{i+1}\in\mathrm L_I^M$ such that $M\models\varphi(\hat v_{i+1},\hat a_{i+1})$. Given any big enough $\mathrm L_\alpha^M\subseteq\mathrm L_I^M$ containing both $\hat a_{i+1}$ and $\hat v_{i+1}$, define $$ \begin{aligned} v_{i+1}&=\min\{ v\in\mathrm L_\alpha^M: M\models\exists x\in a_i\cap\mathrm L_\alpha^M\ \varphi(v,x) \},\ \text{and}\\ a_{i+1}&=\min\{ x\in\mathrm L_\alpha^M:M\models x\in a_i\wedge\varphi(v_{i+1},x) \}, \end{aligned} $$ where the minima are taken with respect to the $\mathrm L^M$-order. These exist because $M$ has $\Delta_{n+1}$-foundation. It can be verified that this choice of $a_{i+1}$ does not depend on the choice of $\alpha$.

Since $\omega^M=\omega$, one sees that $i\mapsto a_i$ is a total function $\omega^M\to M$. The rest is straightforward.