If

Suppose $\kappa$ is $\lambda$-supercompact for some $\lambda \geq \kappa$, and let $j: V \rightarrow M$ is be an elementary embedding with critical point $\kappa$ such that $j(\kappa) > \lambda$ and $M^{\lambda} \subseteq M$ for some inner model $M$, then you can verify M$. First, observe that $V$ and $M$ actually has agree on $j''\lambda^{{<}\kappa}$ so P_{\kappa}\lambda$ because $M$ is closed under ${<}\kappa$ sequences. In particular, this means that $\lambda^{{<}\kappa} \leq (\lambda^{{<}\kappa})^M$ since $M \subseteq V$. But this then means that $j(\kappa) > (\lambda^{{<}\kappa})^M \geq \lambda^{{<}\kappa}$ (because $j(\kappa)$ is inaccessible in $M$ and $j(\kappa) > \kappa$ j(\kappa)$ is greater than both $\lambda$ and $\lambda$) \kappa$. Next, note that any $x \in P_{\kappa}\lambda$ will be a subset of $\lambda$ having size less than the critical point $\kappa$ so that $j(x) = j''x \subseteq j''\lambda$.

[Specifically, if for some $\alpha < \kappa$, we have a bijection $f: \alpha \rightarrow x$, then $j(f)$ will be a bijection between $j(\alpha) = \alpha$ and $j(x)$. So every element of $j(x)$ is of the form $j(f)(\beta)$ for some $\beta < \alpha$, but $j(f)(\beta) = j(f(\beta)) \in j''x$ since $\beta$ is also below the critical point.]

Also, $M$ will actually contain $h = j \upharpoonright \lambda$ by its closure under $\lambda$ sequences. Therefore, $M$ will have $j''P_{\kappa}\lambda = \{j(x)| x \in P_{\kappa}\lambda\} = \{j''x| x \in P_{\kappa}\lambda\} = \{h''x| x \in P_{\kappa}\lambda\}$. Now letting $g: P_{\kappa}\lambda \rightarrow \lambda^{{<}\kappa}$ be a bijection in $V$, we will have a bijection $j(g) \upharpoonright j''P_{\kappa}\lambda: j''P_{\kappa}\lambda \rightarrow j''\lambda^{{<}\kappa}$ in $M$. Therefore, $M$ will have the true collection range of sets (and functions) having hereditary $j(g)$, which is exactly $j''\lambda^{{<}\kappa}$. Now, since $C$ has size at most $\lambda^{{<}\kappa}$. Consequently<}\kappa}$ (in $V$), we may let $e: \lambda^{{<}\kappa} \rightarrow C$ be a surjection. Then $j(e) \upharpoonright j''\lambda^{{<}\kappa}: j''\lambda^{{<}\kappa} \rightarrow j''C$ is a surjection in $M$ so similarly, once you verify its range, $D = j''C$, will be in $M$. But $M$ will also know that $D$ j''\lambda^{{<}\kappa}$ has (hereditary) size at most $\lambda^{{<}\kappa} < j(\kappa)$ in $V$, you do so for because $M$ as wellcan construct $j \upharpoonright \lambda^{{<}\kappa}$ from $j''\lambda^{{<}\kappa}$ by virtue of $j$ being order-preserving. Therefore $\bigcup D \in j(C)$ by the ${<}j(\kappa)$-directed closure of $j(C)$ in $M$, as you mention.

If

Also, if $x \in C$, then $|x| < \kappa$ and $x \subseteq \lambda$ so $j(x) = j''x \subseteq j''\lambda$. The equality holds because given a bijection $f: \alpha \rightarrow x$ for some $\alpha$ below the critical pointTherefore, $j(f)$ will be a bijection between $j(\alpha) \bigcup D = \alpha$ and $j(x)$. So every element of $j(x)$ is of the form $j(f)(\beta)$ for some $\beta < \alpha$, but $j(f)(\beta) = j(f(\beta)) bigcup j''C \in j''x$ since $\beta$ is also below the critical pointsubseteq j''\lambda$.

A

If $\lambda$-supercompactness j: V \rightarrow M$ is an elementary embedding with critical point $\kappa$ is such that $j(\kappa) > \lambda$ and $M^{\lambda} \subseteq M$ for some inner model $M$, then you can verify that $M$ actually a has $\lambda^{{<}\kappa}$-supercompactness embedding meaning j''\lambda^{{<}\kappa}$ so that $j(\kappa) > \lambda^{{<}\kappa}$ (because $j(\kappa)$ is inaccessible in $M$ and $M^{\lambda^{{<}\kappa}} j(\kappa) > \subseteq kappa$ and $\lambda$) and $M$ so will actually contain the true collection of sets (and functions) having hereditary size at most $\lambda^{{<}\kappa}$. Consequently, once you verify that $D$ has (hereditary) size at most $\lambda^{{<}\kappa} < j(\kappa)$ in $V$, you do so for $M$ as well. Therefore $\bigcup D \in j(C)$ by the ${<}j(\kappa)$-directed closure of $j(C)$ in $M$, as you mention.

If $x \in C$, then $|x| < \kappa$ and $x \subseteq \lambda$ so $j(x) = j''x \subseteq j''\lambda$. The equality holds because given a bijection $f: \alpha \rightarrow x$ for some $\alpha$ below the critical point, $j(f)$ will be a bijection between $j(\alpha) = \alpha$ and $j(x)$. So every element of $j(x)$ is of the form $j(f)(\beta)$ for some $\beta < \alpha$, but $j(f)(\beta) = j(f(\beta)) \in j''x$ since $\beta$ is also below the critical point.

A $\lambda$-supercompactness embedding with critical point $\kappa$ is actually a $\lambda^{{<}\kappa}$-supercompactness embedding meaning that $j(\kappa) > \lambda^{{<}\kappa}$ and $M^{\lambda^{{<}\kappa}} \subseteq M$ so once you verify that $D$ has size at most $\lambda^{{<}\kappa} < j(\kappa)$ in $V$, you do so for $M$ as well. Therefore $\bigcup D \in j(C)$ by the ${<}j(\kappa)$-directed closure of $j(C)$ in $M$, as you mention.

If $x \in C$, then $|x| < \kappa$ and $x \subseteq \lambda$ so $j(x) = j''x \subseteq j''\lambda$. The equality holds because given a bijection $f: \alpha \rightarrow x$ for some $\alpha$ below the critical point, $j(f)$ will be a bijection between $j(\alpha) = \alpha$ and $j(x)$. So every element of $j(x)$ is of the form $j(f)(\beta)$ for some $\beta < \alpha$, but $j(f)(\beta) = j(f(\beta)) \in j''x$ since $\beta$ is also below the critical point.